PAPER · REASONING · PRACTICE

Triangle sides, medians and altitudes

Consolidate strict triangle inequalities, perimeter and the definitions of supporting segments.

40 minutes · Year 8, upper volume · text pp. 1–4 · foundation lessons 1–2

Textbook reference: Year 8, upper · Chapter 11: triangles · lessons 1–2

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

Try three paper strips of lengths 2,3,5 and then 2,3,4. Why does equality fail to enclose a triangle?

Organise the key idea

Two shorter sides must total more than the longest; a median bisects the opposite side, while an altitude is perpendicular to its supporting line.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

Which three lengths can form a nondegenerate triangle? Show the shortest-two-sides test.

4,6,9
2,3,5
1,4,6
0,4,4

All three lengths must be positive.
For 4,6,9, the test is 4+6=10>9. Equality would place the segments on one line.
Compare with: 4,6,9

All three lengths must be positive.
For 4,6,9, the test is 4+6=10>9. Equality would place the segments on one line.

Check your work

Choose the statement justified by the stated conditions and explain why.

Guided attempt2

M is the midpoint of BC and BC=12 cm. Explain why AM is a median and find BM.

BM=MC because ___.
AM joins ___ to ___; BM=___ cm.

Geometry diagram; not drawn to scale except for construction tasks.

Start from the definition, not the apparent slope of the line.
A median joins a vertex to the midpoint of its opposite side.
AM joins vertex A to the midpoint M; BM=6 cm.

A median joins a vertex to the midpoint of its opposite side.
BM=MC=BC/2=12/2=6 cm.

Check your work

Name the relevant vertex, opposite side and defining condition.

Prerequisite3

Which requirement is missing from lengths 0,5,5?

Two sides must be equal.
The sides must be integers.
Every side must have positive length.
The perimeter must exceed 20.

A side of length zero collapses two vertices.
The shortest-side sum is also only 0+5=5, not greater than 5.
Compare with: Every side must have positive length.

A side of length zero collapses two vertices.
The shortest-side sum is also only 0+5=5, not greater than 5.

Check your work

Choose the statement justified by the stated conditions and explain why.

Prerequisite4

A triangle has sides 5,5,8 cm. Find its perimeter.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Add all three side lengths.
5+5+8=18 cm; 5+5>8 also confirms feasibility.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice5

Can lengths 3,7,10 form a triangle?

No: 3+7=10, and the inequality must be strict.
Yes: the two sides add to the third.
Yes: all lengths are positive.
Only if it is a right triangle.

Compare the shorter-side sum with the longest side.
3+7=10 gives a straight configuration, not a triangle.
Compare with: No: 3+7=10, and the inequality must be strict.

Compare the shorter-side sum with the longest side.
3+7=10 gives a straight configuration, not a triangle.

Check your work

Choose the statement justified by the stated conditions and explain why.

Independent practice6

An isosceles triangle has equal sides 6 cm and base 9 cm. Find its perimeter and check feasibility.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The sides are 6,6,9, with 6+6>9.
The perimeter is 6+6+9=21 cm.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice7

AM is a median of △ABC and BC=15 cm. Find BM.

Geometry diagram; not drawn to scale except for construction tasks.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The median ends at midpoint M, so BM=MC.
BM=15/2=7.5 cm.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice8

H lies on the line BC extended and AH⊥BC. Is AH an altitude of △ABC even though H is outside segment BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Start from the definition, not the apparent slope of the line.
AH starts at vertex A and is perpendicular to the line BC.
Yes. An altitude meets the line containing the opposite side, not necessarily the segment itself.

AH starts at vertex A and is perpendicular to the line BC.
Those are the defining conditions; H may lie on the extension in an obtuse triangle.

Check your work

Name the relevant vertex, opposite side and defining condition.

Exit question9

Can 4,4,8 cm form a triangle? Give one decisive calculation.

No, because 4+4=8.
Yes, because two sides are equal.
Yes, because its perimeter is 16.
Only if the longest side is the base.

The shortest two sides must sum to more than the longest.
Here equality holds, so the vertices would be collinear.
Compare with: No, because 4+4=8.

The shortest two sides must sum to more than the longest.
Here equality holds, so the vertices would be collinear.

Check your work

Choose the statement justified by the stated conditions and explain why.

Exit question10

A median AM is drawn with M the midpoint of BC. Does “median” alone mean AM⊥BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Start from the definition, not the apparent slope of the line.
A median specifies where its endpoint lies, not the angle it makes.
No. The midpoint condition defines a median; perpendicularity is an additional condition.

A median specifies where its endpoint lies, not the angle it makes.
For A=(1,3), B=(−4,0), C=(4,0), M=(0,0), M is the midpoint but AM is not perpendicular to BC.

Check your work

Name the relevant vertex, opposite side and defining condition.

Further practice11

Two sides are 4 cm and 9 cm. Which offered third side makes a triangle?

13 cm
14 cm
8 cm
5 cm

The third side must be greater than 9−4=5 and less than 9+4=13.
Only 8 is strictly between the limits.
Compare with: 8 cm

The third side must be greater than 9−4=5 and less than 9+4=13.
Only 8 is strictly between the limits.

Check your work

Choose the statement justified by the stated conditions and explain why.

Further practice12

A triangle has sides 2.5,3.5,4 cm. Find its perimeter.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

All lengths are positive and 2.5+3.5>4.
2.5+3.5+4=10 cm.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice13

AM is a median and BC=18 cm. Find CM.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

M is the midpoint, so CM=BM=BC/2.
CM=18/2=9 cm.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice14

In the diagram, which given condition makes AH an altitude from A?

Geometry diagram; not drawn to scale except for construction tasks.

Start from the definition, not the apparent slope of the line.
The opposite side to A is BC.
H lies on line BC and AH⊥BC.

The opposite side to A is BC.
The perpendicular from A to the line BC is an altitude.

Check your work

Name the relevant vertex, opposite side and defining condition.

Extension15

An isosceles triangle has two stated side lengths 4 cm and 9 cm. Which equal-side choice is possible?

Both choices are possible.
Neither choice is possible.
Equal sides 9 cm, base 4 cm.
Equal sides 4 cm, base 9 cm.

If the equal sides are 4, then 4+4<9, so that case fails.
If the equal sides are 9, then 9+4>9; the triangle exists.
Compare with: Equal sides 9 cm, base 4 cm.

If the equal sides are 4, then 4+4<9, so that case fails.
If the equal sides are 9, then 9+4>9; the triangle exists.

Check your work

Choose the statement justified by the stated conditions and explain why.

Extension16

Does a vertex angle bisector always bisect the opposite side? Test the shown unequal-side triangle.

Yes, by the definition of bisector.
Yes, for every triangle with positive sides.
Only the drawing direction matters.
No; angle equality alone does not give a midpoint.

Geometry diagram; not drawn to scale except for construction tasks.

An angle bisector divides an angle, whereas a median divides a side.
In A=(0,3), B=(−4,0), C=(3,0), the side lengths AB=5 and AC=√18 differ; the internal bisector meets BC away from its midpoint.
Compare with: No; angle equality alone does not give a midpoint.

An angle bisector divides an angle, whereas a median divides a side.
In A=(0,3), B=(−4,0), C=(3,0), the side lengths AB=5 and AC=√18 differ; the internal bisector meets BC away from its midpoint.

Check your work

Choose the statement justified by the stated conditions and explain why.

Preparation and assessment notes

Prerequisites

Positive lengths, perimeter, midpoint and perpendicularity.

Look for in students’ work

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

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.

Print edition

Textbook reference and authorship

Chapter-core practice after the mapped textbook lessons; these tasks do not replace every first-teaching period.

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. 1–4 · foundation lessons 1–2

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Triangle sides, medians and altitudes

Year 8, upper volume · text pp. 1–4 · foundation lessons 1–2 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Which three lengths can form a nondegenerate triangle? Show the shortest-two-sides test.

4,6,9
2,3,5
1,4,6
0,4,4

Guided attempt2

M is the midpoint of BC and BC=12 cm. Explain why AM is a median and find BM.

BM=MC because ___.
AM joins ___ to ___; BM=___ cm.

Geometry diagram; not drawn to scale except for construction tasks.

Prerequisite3

Which requirement is missing from lengths 0,5,5?

Two sides must be equal.
The sides must be integers.
Every side must have positive length.
The perimeter must exceed 20.

Prerequisite4

A triangle has sides 5,5,8 cm. Find its perimeter.

Triangle sides, medians and altitudes

Year 8, upper volume · text pp. 1–4 · foundation lessons 1–2 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Can lengths 3,7,10 form a triangle?

No: 3+7=10, and the inequality must be strict.
Yes: the two sides add to the third.
Yes: all lengths are positive.
Only if it is a right triangle.

Independent practice6

An isosceles triangle has equal sides 6 cm and base 9 cm. Find its perimeter and check feasibility.

Independent practice7

AM is a median of △ABC and BC=15 cm. Find BM.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice8

H lies on the line BC extended and AH⊥BC. Is AH an altitude of △ABC even though H is outside segment BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Exit question9

Can 4,4,8 cm form a triangle? Give one decisive calculation.

No, because 4+4=8.
Yes, because two sides are equal.
Yes, because its perimeter is 16.
Only if the longest side is the base.

Exit question10

A median AM is drawn with M the midpoint of BC. Does “median” alone mean AM⊥BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Triangle sides, medians and altitudes — Further practice

Year 8, upper volume · text pp. 1–4 · foundation lessons 1–2

Further practice11

Two sides are 4 cm and 9 cm. Which offered third side makes a triangle?

13 cm
14 cm
8 cm
5 cm

Further practice12

A triangle has sides 2.5,3.5,4 cm. Find its perimeter.

Further practice13

AM is a median and BC=18 cm. Find CM.

Further practice14

In the diagram, which given condition makes AH an altitude from A?

Geometry diagram; not drawn to scale except for construction tasks.

Extension15

An isosceles triangle has two stated side lengths 4 cm and 9 cm. Which equal-side choice is possible?

Both choices are possible.
Neither choice is possible.
Equal sides 9 cm, base 4 cm.
Equal sides 4 cm, base 9 cm.

Extension16

Does a vertex angle bisector always bisect the opposite side? Test the shown unequal-side triangle.

Yes, by the definition of bisector.
Yes, for every triangle with positive sides.
Only the drawing direction matters.
No; angle equality alone does not give a midpoint.

Geometry diagram; not drawn to scale except for construction tasks.

Triangle sides, medians and altitudes — Teacher answers

Year 8, upper volume · text pp. 1–4 · foundation lessons 1–2

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Worked example1

Which three lengths can form a nondegenerate triangle? Show the shortest-two-sides test.

4,6,9
2,3,5
1,4,6
0,4,4

Reference answer

4,6,9

All three lengths must be positive.
For 4,6,9, the test is 4+6=10>9. Equality would place the segments on one line.

Choose the statement justified by the stated conditions and explain why.

Guided attempt2

M is the midpoint of BC and BC=12 cm. Explain why AM is a median and find BM.

BM=MC because ___.
AM joins ___ to ___; BM=___ cm.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

AM joins vertex A to the midpoint M; BM=6 cm.

A median joins a vertex to the midpoint of its opposite side.
BM=MC=BC/2=12/2=6 cm.

Name the relevant vertex, opposite side and defining condition.

Prerequisite3

Which requirement is missing from lengths 0,5,5?

Two sides must be equal.
The sides must be integers.
Every side must have positive length.
The perimeter must exceed 20.

Reference answer

Every side must have positive length.

A side of length zero collapses two vertices.
The shortest-side sum is also only 0+5=5, not greater than 5.

Choose the statement justified by the stated conditions and explain why.

Prerequisite4

A triangle has sides 5,5,8 cm. Find its perimeter.

Reference answer

18 cm.

5+5+8

Add all three side lengths.
5+5+8=18 cm; 5+5>8 also confirms feasibility.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice5

Can lengths 3,7,10 form a triangle?

No: 3+7=10, and the inequality must be strict.
Yes: the two sides add to the third.
Yes: all lengths are positive.
Only if it is a right triangle.

Reference answer

No: 3+7=10, and the inequality must be strict.

Compare the shorter-side sum with the longest side.
3+7=10 gives a straight configuration, not a triangle.

Choose the statement justified by the stated conditions and explain why.

Independent practice6

An isosceles triangle has equal sides 6 cm and base 9 cm. Find its perimeter and check feasibility.

Reference answer

21 cm.

6+6+9

The sides are 6,6,9, with 6+6>9.
The perimeter is 6+6+9=21 cm.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice7

AM is a median of △ABC and BC=15 cm. Find BM.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

7.5 cm.

15/2

The median ends at midpoint M, so BM=MC.
BM=15/2=7.5 cm.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice8

H lies on the line BC extended and AH⊥BC. Is AH an altitude of △ABC even though H is outside segment BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Yes. An altitude meets the line containing the opposite side, not necessarily the segment itself.

AH starts at vertex A and is perpendicular to the line BC.
Those are the defining conditions; H may lie on the extension in an obtuse triangle.

Name the relevant vertex, opposite side and defining condition.

Exit question9

Can 4,4,8 cm form a triangle? Give one decisive calculation.

No, because 4+4=8.
Yes, because two sides are equal.
Yes, because its perimeter is 16.
Only if the longest side is the base.

Reference answer

No, because 4+4=8.

The shortest two sides must sum to more than the longest.
Here equality holds, so the vertices would be collinear.

Choose the statement justified by the stated conditions and explain why.

Exit question10

A median AM is drawn with M the midpoint of BC. Does “median” alone mean AM⊥BC? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No. The midpoint condition defines a median; perpendicularity is an additional condition.

A median specifies where its endpoint lies, not the angle it makes.
For A=(1,3), B=(−4,0), C=(4,0), M=(0,0), M is the midpoint but AM is not perpendicular to BC.

Name the relevant vertex, opposite side and defining condition.

Further practice11

Two sides are 4 cm and 9 cm. Which offered third side makes a triangle?

13 cm
14 cm
8 cm
5 cm

Reference answer

8 cm

The third side must be greater than 9−4=5 and less than 9+4=13.
Only 8 is strictly between the limits.

Choose the statement justified by the stated conditions and explain why.

Further practice12

A triangle has sides 2.5,3.5,4 cm. Find its perimeter.

Reference answer

10 cm.

2.5+3.5+4

All lengths are positive and 2.5+3.5>4.
2.5+3.5+4=10 cm.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice13

AM is a median and BC=18 cm. Find CM.

Reference answer

9 cm.

18/2

M is the midpoint, so CM=BM=BC/2.
CM=18/2=9 cm.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice14

In the diagram, which given condition makes AH an altitude from A?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

H lies on line BC and AH⊥BC.

The opposite side to A is BC.
The perpendicular from A to the line BC is an altitude.

Name the relevant vertex, opposite side and defining condition.

Extension15

An isosceles triangle has two stated side lengths 4 cm and 9 cm. Which equal-side choice is possible?

Both choices are possible.
Neither choice is possible.
Equal sides 9 cm, base 4 cm.
Equal sides 4 cm, base 9 cm.

Reference answer

Equal sides 9 cm, base 4 cm.

If the equal sides are 4, then 4+4<9, so that case fails.
If the equal sides are 9, then 9+4>9; the triangle exists.

Choose the statement justified by the stated conditions and explain why.

Extension16

Does a vertex angle bisector always bisect the opposite side? Test the shown unequal-side triangle.

Yes, by the definition of bisector.
Yes, for every triangle with positive sides.
Only the drawing direction matters.
No; angle equality alone does not give a midpoint.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No; angle equality alone does not give a midpoint.

An angle bisector divides an angle, whereas a median divides a side.
In A=(0,3), B=(−4,0), C=(3,0), the side lengths AB=5 and AC=√18 differ; the internal bisector meets BC away from its midpoint.

Choose the statement justified by the stated conditions and explain why.