Compare joining a corresponding point pair with constructing its perpendicular bisector. Which line actually exchanges the endpoints?
Organise the key idea
The axis exchanging two distinct points is their perpendicular bisector. For several pairs, the same line must work for every pair; a figure’s axis must also map the whole figure onto itself.
Use the lesson question here for paper discussion; the original activity opens separately.
Worked example1
A and B are a pair of corresponding points. Construct their symmetry axis with ruler and compass. Keep the arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
The symmetry axis of a pair of distinct corresponding points is their perpendicular bisector.
Use the same compass radius from both endpoints, large enough for two arc intersections.
Join the two arc intersections, not the original endpoints.
Choose one compass radius greater than half the drawn segment AB.
With centres A and B, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of AB.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Reference answer
The full line through the midpoint of AB, perpendicular to AB. Construct it with two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Guided attempt2
C and D are distinct corresponding points on the slanted segment. Complete the construction using the scaffold, without estimating the axis by eye.
Choose radius greater than ___ of CD.
Use centres ___ and ___ with the same radius to obtain X and Y.
Draw line ___; check it is ___ to CD and passes through its ___.
Geometry diagram; not drawn to scale except for construction tasks.
A slanted segment uses the same construction as a horizontal one.
The radius must remain unchanged between the two centres.
The axis joins the arc intersections X and Y.
Take a common radius greater than half CD.
Draw arcs from C and D on both sides, meeting at X and Y.
Line XY passes through the midpoint and is perpendicular to CD.
Extend the line and leave the arcs as evidence of the construction.
Reference answer
The perpendicular bisector of CD, formed by the two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Use equal-radius arcs and draw the perpendicular bisector as a full line.
Keep traces rather than relying on apparent symmetry.
Prerequisite3
Which two checks certify that a line is the axis exchanging distinct A and A′?
It is perpendicular to AA′ somewhere.
It only passes through the midpoint.
It passes through the midpoint of AA′ and is perpendicular to AA′.
It passes through A and A′.
Recall the complete definition of perpendicular bisector.
One condition concerns position; the other concerns angle.
Require both the midpoint and a right angle.
A reflection axis bisects the segment joining distinct corresponding points.
It is also perpendicular to that joining segment.
Reference answer
Both midpoint and perpendicularity are required.
Check your work
State both conditions together.
Prerequisite4
To obtain two distinct intersections of equal-radius circles centred at A and B, how should the radius compare with half AB?
It must equal half AB.
It must be smaller than half AB.
It must be zero.
It must be greater than half AB.
Imagine two circles that are just touching.
Just touching supplies one point, not two.
Open the compass beyond half the endpoint separation.
With equal radii r, two intersections require AB<2r.
Hence r>AB/2. At equality the circles touch at the midpoint.
Reference answer
Greater than half AB; equality gives only one touching point.
Check your work
Use a strict greater-than condition and explain why equality is insufficient.
Independent practice5
E and F are corresponding points. Construct their full symmetry axis with ruler and compass.
Geometry diagram; not drawn to scale except for construction tasks.
The symmetry axis of a pair of distinct corresponding points is their perpendicular bisector.
Use the same compass radius from both endpoints, large enough for two arc intersections.
Join the two arc intersections, not the original endpoints.
Choose one compass radius greater than half the drawn segment EF.
With centres E and F, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of EF.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Reference answer
The full line through the midpoint of EF, perpendicular to EF. Construct it with two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Independent practice6
G and H are corresponding points on the vertical segment. Construct their symmetry axis and mark the right angle.
Geometry diagram; not drawn to scale except for construction tasks.
The symmetry axis of a pair of distinct corresponding points is their perpendicular bisector.
Use the same compass radius from both endpoints, large enough for two arc intersections.
Join the two arc intersections, not the original endpoints.
Choose one compass radius greater than half the drawn segment GH.
With centres G and H, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of GH.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Reference answer
The full line through the midpoint of GH, perpendicular to GH. Construct it with two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Independent practice7
A corresponds to A′ and B corresponds to B′. Construct one line that exchanges both pairs, and show how the second pair checks your first construction.
Geometry diagram; not drawn to scale except for construction tasks.
Start with one distinct corresponding pair.
One line must satisfy the midpoint and right-angle checks for both pairs.
The second pair’s perpendicular bisector should coincide with the first.
Construct the perpendicular bisector of AA′ with equal-radius arcs.
Check that the line also passes through the midpoint of BB′ and is perpendicular to BB′.
Alternatively construct the second perpendicular bisector; it coincides with the first.
Extend this one common line; do not count it twice.
Reference answer
The common perpendicular bisector of AA′ and BB′. The two constructed perpendicular bisectors coincide.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Construct a valid axis and explicitly verify the other pair.
Independent practice8
In the non-equilateral isosceles triangle, AB=AC. Use ruler and compass to construct the symmetry axis. Explain why the line you construct passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Which two vertices must exchange under the axis?
Construct the perpendicular bisector of the base BC.
Use AB=AC and the perpendicular-bisector converse to locate A.
Use equal-radius arcs from B and C to construct the perpendicular bisector of BC.
AB=AC means A is equidistant from B and C, so A lies on the constructed line.
Reflection fixes A and exchanges B and C; the entire triangle matches.
Reference answer
Construct the perpendicular bisector of BC. Since AB=AC, the converse puts A on that line; it is the triangle’s only axis.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Retain compass traces for the base perpendicular bisector.
Use equal endpoint distances to justify that the axis passes through A.
Exit question9
A pupil joins distinct corresponding points A and A′ and labels the joining line “axis”. Explain the error and state the line that should be constructed.
Geometry diagram; not drawn to scale except for construction tasks.
Ask what reflection does to a point on the axis.
A and A′ must exchange, not both stay fixed.
Use the perpendicular bisector of AA′.
Every point on a reflection axis remains fixed.
A and A′ are distinct points to be exchanged, so their joining line cannot be the axis.
The correct line passes through their midpoint at a right angle.
Reference answer
The joining line would fix A and A′ instead of exchanging them. Construct its perpendicular bisector.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Explain the fixed-point error and name the perpendicular bisector.
Exit question10
J and J′ are a corresponding pair. Draw their full axis, and mark both the midpoint and right-angle checks. Keep any compass traces used.
Geometry diagram; not drawn to scale except for construction tasks.
Use the same construction as in the worked example.
An axis needs both the midpoint and a right angle.
Join the equal-radius arc intersections and extend the line.
Construct or verify the midpoint of JJ′ with equal-radius arcs.
The line through the two arc intersections is perpendicular to JJ′.
Extend the line and mark both required checks.
Reference answer
The line through the midpoint of JJ′ perpendicular to JJ′, extended in both directions.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show a full perpendicular bisector with midpoint and right-angle evidence.
Further practice11
K and L are corresponding points on a slanted segment. Construct the axis without measuring its angle.
Geometry diagram; not drawn to scale except for construction tasks.
The symmetry axis of a pair of distinct corresponding points is their perpendicular bisector.
Use the same compass radius from both endpoints, large enough for two arc intersections.
Join the two arc intersections, not the original endpoints.
Choose one compass radius greater than half the drawn segment KL.
With centres K and L, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of KL.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Reference answer
The full line through the midpoint of KL, perpendicular to KL. Construct it with two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Further practice12
R and S are corresponding points. Construct the axis, preserving the same compass radius from both centres.
Geometry diagram; not drawn to scale except for construction tasks.
The symmetry axis of a pair of distinct corresponding points is their perpendicular bisector.
Use the same compass radius from both endpoints, large enough for two arc intersections.
Join the two arc intersections, not the original endpoints.
Choose one compass radius greater than half the drawn segment RS.
With centres R and S, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of RS.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Reference answer
The full line through the midpoint of RS, perpendicular to RS. Construct it with two equal-radius arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Further practice13
Use ruler and compass to draw all axes of a square. Include both side-midpoint axes and vertex axes.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Two lines through opposite side midpoints and the two diagonals.
Construct the perpendicular bisectors of two adjacent sides with equal-radius arcs.
Draw the two diagonals using the ruler.
Extend the four different lines and verify the whole-square reflection test.
Reference answer
Four lines: the perpendicular bisectors of two adjacent sides and the two diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Retain traces for the perpendicular-bisector constructions.
Further practice14
Use ruler and compass to construct all axes of a non-square rectangle. Give a reason for leaving out the diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
Construct the perpendicular bisectors of two adjacent sides.
They pass through the rectangle centre and the opposite side midpoints.
Check the two full axes; reject diagonal candidates using unequal adjacent sides.
Reference answer
Two lines: the perpendicular bisectors of two adjacent sides. Diagonal folds fail because the adjacent side lengths differ.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Show compass traces for both midpoint axes.
Extension15
OA and OB are two rays forming a right angle. Construct the symmetry axis of this two-ray figure using ruler and compass, and explain which ray each side reflects to.
Geometry diagram; not drawn to scale except for construction tasks.
The two sides must exchange while the vertex stays fixed.
Recall the ruler-and-compass construction of an angle bisector.
Choose E inside the angle and extend OE as a full symmetry axis.
Draw an arc centred at O meeting the rays at C and D, so OC=OD.
Use equal-radius arcs centred at C and D to obtain an intersection E inside the angle, different from O.
Draw the full line OE; the equal-side construction bisects the angle.
Reflection in this line fixes O and exchanges the two rays.
Reference answer
The full line through O along the internal angle bisector; it exchanges OA and OB.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Use equal-radius arc construction and identify the internal-bisector direction.
Draw a full axis and explain the exchanged rays.
Extension16
A corresponds to A′ and B corresponds to B′. Can one line exchange both supplied pairs? Use ruler and compass to compare their required axes, rather than choosing a line halfway between them.
Geometry diagram; not drawn to scale except for construction tasks.
Each distinct corresponding pair determines a unique perpendicular bisector.
Construct both required lines and compare their positions.
They are different parallel lines, so they cannot be one common axis.
Construct the perpendicular bisector of AA′; it is the unique axis for that distinct pair.
Construct the perpendicular bisector of BB′.
The two lines are parallel and do not coincide, so no single line works for both pairs.
An averaged line would fail at least one pair and is not a substitute.
Reference answer
No. The two required perpendicular bisectors are parallel but distinct; a common axis cannot satisfy both pairs.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
Construct or identify both perpendicular bisectors.
Explain why distinct required axes rule out a common axis.
Preparation and assessment notes
Prerequisites
The perpendicular-bisector property, corresponding points, equal-radius compass arcs and full straight lines. Coordinates are internal diagram data, not a required method.
Look for in students’ work
Use a ruler and compass; keep arcs visible. Show the solution axis only in the teacher answer. Check midpoint and perpendicularity separately. This lesson constructs axes, not later coordinate rules or whole reflected polygons.
Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.
All reference answers
Worked example1
A and B are a pair of corresponding points. Construct their symmetry axis with ruler and compass. Keep the arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of AB, perpendicular to AB. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment AB.
With centres A and B, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of AB.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2
C and D are distinct corresponding points on the slanted segment. Complete the construction using the scaffold, without estimating the axis by eye.
Choose radius greater than ___ of CD.
Use centres ___ and ___ with the same radius to obtain X and Y.
Draw line ___; check it is ___ to CD and passes through its ___.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The perpendicular bisector of CD, formed by the two equal-radius arc intersections.
Take a common radius greater than half CD.
Draw arcs from C and D on both sides, meeting at X and Y.
Line XY passes through the midpoint and is perpendicular to CD.
Extend the line and leave the arcs as evidence of the construction.
Use equal-radius arcs and draw the perpendicular bisector as a full line.
Keep traces rather than relying on apparent symmetry.
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3
Which two checks certify that a line is the axis exchanging distinct A and A′?
It is perpendicular to AA′ somewhere.
It only passes through the midpoint.
It passes through the midpoint of AA′ and is perpendicular to AA′.
It passes through A and A′.
Reference answer
Both midpoint and perpendicularity are required.
A reflection axis bisects the segment joining distinct corresponding points.
It is also perpendicular to that joining segment.
State both conditions together.
Prerequisite4
To obtain two distinct intersections of equal-radius circles centred at A and B, how should the radius compare with half AB?
It must equal half AB.
It must be smaller than half AB.
It must be zero.
It must be greater than half AB.
Reference answer
Greater than half AB; equality gives only one touching point.
With equal radii r, two intersections require AB<2r.
Hence r>AB/2. At equality the circles touch at the midpoint.
Use a strict greater-than condition and explain why equality is insufficient.
Independent practice5
E and F are corresponding points. Construct their full symmetry axis with ruler and compass.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of EF, perpendicular to EF. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment EF.
With centres E and F, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of EF.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6
G and H are corresponding points on the vertical segment. Construct their symmetry axis and mark the right angle.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of GH, perpendicular to GH. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment GH.
With centres G and H, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of GH.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7
A corresponds to A′ and B corresponds to B′. Construct one line that exchanges both pairs, and show how the second pair checks your first construction.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The common perpendicular bisector of AA′ and BB′. The two constructed perpendicular bisectors coincide.
Construct the perpendicular bisector of AA′ with equal-radius arcs.
Check that the line also passes through the midpoint of BB′ and is perpendicular to BB′.
Alternatively construct the second perpendicular bisector; it coincides with the first.
Extend this one common line; do not count it twice.
Construct a valid axis and explicitly verify the other pair.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8
In the non-equilateral isosceles triangle, AB=AC. Use ruler and compass to construct the symmetry axis. Explain why the line you construct passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Construct the perpendicular bisector of BC. Since AB=AC, the converse puts A on that line; it is the triangle’s only axis.
Use equal-radius arcs from B and C to construct the perpendicular bisector of BC.
AB=AC means A is equidistant from B and C, so A lies on the constructed line.
Reflection fixes A and exchanges B and C; the entire triangle matches.
Retain compass traces for the base perpendicular bisector.
Use equal endpoint distances to justify that the axis passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9
A pupil joins distinct corresponding points A and A′ and labels the joining line “axis”. Explain the error and state the line that should be constructed.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The joining line would fix A and A′ instead of exchanging them. Construct its perpendicular bisector.
Every point on a reflection axis remains fixed.
A and A′ are distinct points to be exchanged, so their joining line cannot be the axis.
The correct line passes through their midpoint at a right angle.
Explain the fixed-point error and name the perpendicular bisector.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10
J and J′ are a corresponding pair. Draw their full axis, and mark both the midpoint and right-angle checks. Keep any compass traces used.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The line through the midpoint of JJ′ perpendicular to JJ′, extended in both directions.
Construct or verify the midpoint of JJ′ with equal-radius arcs.
The line through the two arc intersections is perpendicular to JJ′.
Extend the line and mark both required checks.
Show a full perpendicular bisector with midpoint and right-angle evidence.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice11
K and L are corresponding points on a slanted segment. Construct the axis without measuring its angle.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of KL, perpendicular to KL. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment KL.
With centres K and L, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of KL.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12
R and S are corresponding points. Construct the axis, preserving the same compass radius from both centres.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of RS, perpendicular to RS. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment RS.
With centres R and S, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of RS.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13
Use ruler and compass to draw all axes of a square. Include both side-midpoint axes and vertex axes.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Four lines: the perpendicular bisectors of two adjacent sides and the two diagonals.
Construct the perpendicular bisectors of two adjacent sides with equal-radius arcs.
Draw the two diagonals using the ruler.
Extend the four different lines and verify the whole-square reflection test.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Retain traces for the perpendicular-bisector constructions.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14
Use ruler and compass to construct all axes of a non-square rectangle. Give a reason for leaving out the diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Two lines: the perpendicular bisectors of two adjacent sides. Diagonal folds fail because the adjacent side lengths differ.
Construct the perpendicular bisectors of two adjacent sides.
They pass through the rectangle centre and the opposite side midpoints.
Check the two full axes; reject diagonal candidates using unequal adjacent sides.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Show compass traces for both midpoint axes.
Geometry diagram; not drawn to scale except for construction tasks.
Extension15
OA and OB are two rays forming a right angle. Construct the symmetry axis of this two-ray figure using ruler and compass, and explain which ray each side reflects to.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through O along the internal angle bisector; it exchanges OA and OB.
Draw an arc centred at O meeting the rays at C and D, so OC=OD.
Use equal-radius arcs centred at C and D to obtain an intersection E inside the angle, different from O.
Draw the full line OE; the equal-side construction bisects the angle.
Reflection in this line fixes O and exchanges the two rays.
Use equal-radius arc construction and identify the internal-bisector direction.
Draw a full axis and explain the exchanged rays.
Geometry diagram; not drawn to scale except for construction tasks.
Extension16
A corresponds to A′ and B corresponds to B′. Can one line exchange both supplied pairs? Use ruler and compass to compare their required axes, rather than choosing a line halfway between them.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
No. The two required perpendicular bisectors are parallel but distinct; a common axis cannot satisfy both pairs.
Construct the perpendicular bisector of AA′; it is the unique axis for that distinct pair.
Construct the perpendicular bisector of BB′.
The two lines are parallel and do not coincide, so no single line works for both pairs.
An averaged line would fail at least one pair and is not a substitute.
Construct or identify both perpendicular bisectors.
Explain why distinct required axes rule out a common axis.
Geometry diagram; not drawn to scale except for construction tasks.
All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.
Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.
Textbook reference and authorship
Original practice aligned to the referenced skills. Use your own textbook for the cited pages.
大湾区初中数学三维课堂（八年级上册）正文.pdf
Year 8, upper volume · text pp. 33–34 · foundation lesson 17
Original questions and explanations; the source textbook and answer PDFs are not reproduced here.
Constructing symmetry axes with ruler and compass
Year 8, upper volume · text pp. 33–34 · foundation lesson 17 · 1/2
Name: __________ Class: ______ Date: __________
Worked example1
A and B are a pair of corresponding points. Construct their symmetry axis with ruler and compass. Keep the arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2
C and D are distinct corresponding points on the slanted segment. Complete the construction using the scaffold, without estimating the axis by eye.
Choose radius greater than ___ of CD.
Use centres ___ and ___ with the same radius to obtain X and Y.
Draw line ___; check it is ___ to CD and passes through its ___.
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3
Which two checks certify that a line is the axis exchanging distinct A and A′?
It is perpendicular to AA′ somewhere.
It only passes through the midpoint.
It passes through the midpoint of AA′ and is perpendicular to AA′.
It passes through A and A′.
Prerequisite4
To obtain two distinct intersections of equal-radius circles centred at A and B, how should the radius compare with half AB?
It must equal half AB.
It must be smaller than half AB.
It must be zero.
It must be greater than half AB.
Constructing symmetry axes with ruler and compass
Year 8, upper volume · text pp. 33–34 · foundation lesson 17 · 2/2
Name: __________ Class: ______ Date: __________
Independent practice5
E and F are corresponding points. Construct their full symmetry axis with ruler and compass.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6
G and H are corresponding points on the vertical segment. Construct their symmetry axis and mark the right angle.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7
A corresponds to A′ and B corresponds to B′. Construct one line that exchanges both pairs, and show how the second pair checks your first construction.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8
In the non-equilateral isosceles triangle, AB=AC. Use ruler and compass to construct the symmetry axis. Explain why the line you construct passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9
A pupil joins distinct corresponding points A and A′ and labels the joining line “axis”. Explain the error and state the line that should be constructed.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10
J and J′ are a corresponding pair. Draw their full axis, and mark both the midpoint and right-angle checks. Keep any compass traces used.
Geometry diagram; not drawn to scale except for construction tasks.
Constructing symmetry axes with ruler and compass — Further practice
Year 8, upper volume · text pp. 33–34 · foundation lesson 17
Further practice11
K and L are corresponding points on a slanted segment. Construct the axis without measuring its angle.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12
R and S are corresponding points. Construct the axis, preserving the same compass radius from both centres.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13
Use ruler and compass to draw all axes of a square. Include both side-midpoint axes and vertex axes.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14
Use ruler and compass to construct all axes of a non-square rectangle. Give a reason for leaving out the diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Extension15
OA and OB are two rays forming a right angle. Construct the symmetry axis of this two-ray figure using ruler and compass, and explain which ray each side reflects to.
Geometry diagram; not drawn to scale except for construction tasks.
Extension16
A corresponds to A′ and B corresponds to B′. Can one line exchange both supplied pairs? Use ruler and compass to compare their required axes, rather than choosing a line halfway between them.
Geometry diagram; not drawn to scale except for construction tasks.
Constructing symmetry axes with ruler and compass — Teacher answers
Year 8, upper volume · text pp. 33–34 · foundation lesson 17
Use a ruler and compass; keep arcs visible. Show the solution axis only in the teacher answer. Check midpoint and perpendicularity separately. This lesson constructs axes, not later coordinate rules or whole reflected polygons.
Worked example1
A and B are a pair of corresponding points. Construct their symmetry axis with ruler and compass. Keep the arc intersections.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of AB, perpendicular to AB. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment AB.
With centres A and B, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of AB.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2
C and D are distinct corresponding points on the slanted segment. Complete the construction using the scaffold, without estimating the axis by eye.
Choose radius greater than ___ of CD.
Use centres ___ and ___ with the same radius to obtain X and Y.
Draw line ___; check it is ___ to CD and passes through its ___.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The perpendicular bisector of CD, formed by the two equal-radius arc intersections.
Take a common radius greater than half CD.
Draw arcs from C and D on both sides, meeting at X and Y.
Line XY passes through the midpoint and is perpendicular to CD.
Extend the line and leave the arcs as evidence of the construction.
Use equal-radius arcs and draw the perpendicular bisector as a full line.
Keep traces rather than relying on apparent symmetry.
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3
Which two checks certify that a line is the axis exchanging distinct A and A′?
It is perpendicular to AA′ somewhere.
It only passes through the midpoint.
It passes through the midpoint of AA′ and is perpendicular to AA′.
It passes through A and A′.
Reference answer
Both midpoint and perpendicularity are required.
A reflection axis bisects the segment joining distinct corresponding points.
It is also perpendicular to that joining segment.
State both conditions together.
Prerequisite4
To obtain two distinct intersections of equal-radius circles centred at A and B, how should the radius compare with half AB?
It must equal half AB.
It must be smaller than half AB.
It must be zero.
It must be greater than half AB.
Reference answer
Greater than half AB; equality gives only one touching point.
With equal radii r, two intersections require AB<2r.
Hence r>AB/2. At equality the circles touch at the midpoint.
Use a strict greater-than condition and explain why equality is insufficient.
Independent practice5
E and F are corresponding points. Construct their full symmetry axis with ruler and compass.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of EF, perpendicular to EF. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment EF.
With centres E and F, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of EF.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6
G and H are corresponding points on the vertical segment. Construct their symmetry axis and mark the right angle.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of GH, perpendicular to GH. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment GH.
With centres G and H, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of GH.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7
A corresponds to A′ and B corresponds to B′. Construct one line that exchanges both pairs, and show how the second pair checks your first construction.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The common perpendicular bisector of AA′ and BB′. The two constructed perpendicular bisectors coincide.
Construct the perpendicular bisector of AA′ with equal-radius arcs.
Check that the line also passes through the midpoint of BB′ and is perpendicular to BB′.
Alternatively construct the second perpendicular bisector; it coincides with the first.
Extend this one common line; do not count it twice.
Construct a valid axis and explicitly verify the other pair.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8
In the non-equilateral isosceles triangle, AB=AC. Use ruler and compass to construct the symmetry axis. Explain why the line you construct passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Construct the perpendicular bisector of BC. Since AB=AC, the converse puts A on that line; it is the triangle’s only axis.
Use equal-radius arcs from B and C to construct the perpendicular bisector of BC.
AB=AC means A is equidistant from B and C, so A lies on the constructed line.
Reflection fixes A and exchanges B and C; the entire triangle matches.
Retain compass traces for the base perpendicular bisector.
Use equal endpoint distances to justify that the axis passes through A.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9
A pupil joins distinct corresponding points A and A′ and labels the joining line “axis”. Explain the error and state the line that should be constructed.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The joining line would fix A and A′ instead of exchanging them. Construct its perpendicular bisector.
Every point on a reflection axis remains fixed.
A and A′ are distinct points to be exchanged, so their joining line cannot be the axis.
The correct line passes through their midpoint at a right angle.
Explain the fixed-point error and name the perpendicular bisector.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10
J and J′ are a corresponding pair. Draw their full axis, and mark both the midpoint and right-angle checks. Keep any compass traces used.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The line through the midpoint of JJ′ perpendicular to JJ′, extended in both directions.
Construct or verify the midpoint of JJ′ with equal-radius arcs.
The line through the two arc intersections is perpendicular to JJ′.
Extend the line and mark both required checks.
Show a full perpendicular bisector with midpoint and right-angle evidence.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice11
K and L are corresponding points on a slanted segment. Construct the axis without measuring its angle.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of KL, perpendicular to KL. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment KL.
With centres K and L, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of KL.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12
R and S are corresponding points. Construct the axis, preserving the same compass radius from both centres.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through the midpoint of RS, perpendicular to RS. Construct it with two equal-radius arc intersections.
Choose one compass radius greater than half the drawn segment RS.
With centres R and S, draw arcs on both sides to obtain two distinct intersections X and Y.
Draw the full line XY. It is the perpendicular bisector of RS.
Mark the segment midpoint and the right angle, then check that the two endpoints exchange under reflection.
Show equal-radius arcs with two distinct intersections.
Draw a full line through the midpoint and perpendicular to the segment.
Retain construction traces and check the corresponding endpoints.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13
Use ruler and compass to draw all axes of a square. Include both side-midpoint axes and vertex axes.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Four lines: the perpendicular bisectors of two adjacent sides and the two diagonals.
Construct the perpendicular bisectors of two adjacent sides with equal-radius arcs.
Draw the two diagonals using the ruler.
Extend the four different lines and verify the whole-square reflection test.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Retain traces for the perpendicular-bisector constructions.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14
Use ruler and compass to construct all axes of a non-square rectangle. Give a reason for leaving out the diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Two lines: the perpendicular bisectors of two adjacent sides. Diagonal folds fail because the adjacent side lengths differ.
Construct the perpendicular bisectors of two adjacent sides.
They pass through the rectangle centre and the opposite side midpoints.
Check the two full axes; reject diagonal candidates using unequal adjacent sides.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Show compass traces for both midpoint axes.
Geometry diagram; not drawn to scale except for construction tasks.
Extension15
OA and OB are two rays forming a right angle. Construct the symmetry axis of this two-ray figure using ruler and compass, and explain which ray each side reflects to.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The full line through O along the internal angle bisector; it exchanges OA and OB.
Draw an arc centred at O meeting the rays at C and D, so OC=OD.
Use equal-radius arcs centred at C and D to obtain an intersection E inside the angle, different from O.
Draw the full line OE; the equal-side construction bisects the angle.
Reflection in this line fixes O and exchanges the two rays.
Use equal-radius arc construction and identify the internal-bisector direction.
Draw a full axis and explain the exchanged rays.
Geometry diagram; not drawn to scale except for construction tasks.
Extension16
A corresponds to A′ and B corresponds to B′. Can one line exchange both supplied pairs? Use ruler and compass to compare their required axes, rather than choosing a line halfway between them.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
No. The two required perpendicular bisectors are parallel but distinct; a common axis cannot satisfy both pairs.
Construct the perpendicular bisector of AA′; it is the unique axis for that distinct pair.
Construct the perpendicular bisector of BB′.
The two lines are parallel and do not coincide, so no single line works for both pairs.
An averaged line would fail at least one pair and is not a substitute.
Construct or identify both perpendicular bisectors.
Explain why distinct required axes rule out a common axis.
Geometry diagram; not drawn to scale except for construction tasks.
