PAPER · REASONING · PRACTICE

Rectangles and rhombi: diagonal conditions and converses — Teacher reference

Keep the parallelogram premise, distinguish equal from perpendicular diagonals, and use the right-triangle median theorem.

40 minutes · Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22

Textbook reference: Year 8, lower volume · Chapter 18: Parallelograms · lessons 19–22

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

Student classroom page 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

3 min: Predict which diagonal properties a non-square rectangle and a non-square rhombus share. 8 min: On paper compare a 4-by-3 rectangle (diagonals 5) with the rhombus (0,0),(3,4),(6,0),(3,−4) (diagonals 6 and 8). Record bisection, equal lengths and perpendicularity separately. Use the stated shapes as premises, not their apparent outlines. 5 min: Identify the shared bisection property and the differing equal/perpendicular properties. Keep the parallelogram premise when using diagonal criteria. 2 min: Individually judge whether perpendicular diagonals alone force an arbitrary quadrilateral to be a rhombus; give a missing condition.

Organise the key idea

A rectangle is a parallelogram with a right angle; a rhombus is a parallelogram with all sides equal. Their special diagonal properties have different roles.

Open the original chapter

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

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

ABCD is a rectangle with AC=10 cm. Find BD and name the special rectangle property.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The diagonals of a rectangle are equal.
BD=AC=10 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

ABCD is a rhombus with AB=5 cm. Find CD and explain why no diagonal calculation is needed.

A rhombus has ____ equal sides.
$AB=BC=CD=DA=\square$ cm.
Therefore CD=____ cm; the information needed is a side length, not a diagonal length.

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

AB and CD are sides; the question does not ask for a diagonal.
Use the rhombus property that all four sides are equal.
CD must have the same stated length as AB, namely 5 cm.

Every side of a rhombus is equal.
CD=AB=5 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

A parallelogram has one right angle. Which special type must it be?

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use the one-right-angle criterion with the parallelogram premise.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

A parallelogram has AB=BC. Which special type follows?

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Equal adjacent sides plus equal opposite sides give all four sides equal.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

ABCD is a parallelogram and AC=BD. State the strongest classification justified by these conditions.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use equal diagonals with the known parallelogram.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

ABCD is a parallelogram and AC⊥BD. Which special type follows? Explain the required premise.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

A general quadrilateral with perpendicular diagonals need not have equal sides.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

∠A=90° in △ABC, BC=10 cm, and M is the midpoint of BC. Find AM.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The median to the hypotenuse of a right triangle equals half the hypotenuse.
AM=10/2=5 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A rhombus has diagonals AC=6 cm and BD=8 cm, meeting at O. Find its side AB, explaining both diagonal properties used.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

AO=3 and BO=4, with ∠AOB=90°.
AB²=3²+4²=25, so AB=5 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Is every rectangle a parallelogram? Explain using the inclusive definition.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Its opposite sides remain parallel even with the extra right-angle condition.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Must every rhombus be a rectangle? Use the shown non-square rhombus.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

This rhombus has perpendicular diagonals but unequal diagonal lengths.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

A rectangle has AC=10 cm and diagonals meeting at O. Find BO.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

BD=AC=10 and O bisects BD.
BO=5 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

A general quadrilateral has equal diagonals. Must it be a rectangle? Explain with the shown trapezium.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The equal-diagonal isosceles trapezium has only one pair of parallel opposite sides.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

A general quadrilateral has perpendicular diagonals. Must it be a rhombus? Give a reason from the shown kite.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Perpendicularity alone does not guarantee diagonal bisection or four equal sides.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

In right △ABC, M is the midpoint of hypotenuse BC and AM=10 cm. Find BC.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

AM=BC/2, so BC=20 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

A rhombus has diagonals 6 cm and 8 cm. Derive its area by splitting it into four right triangles.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The diagonals are perpendicular and bisect each other.
Total area=4×(3×4/2)=24 cm².

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

In a nondegenerate triangle ABC, M is the midpoint of BC and AM=BM. Show that ∠BAC is a right angle, using angles in the two isosceles triangles.

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

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

M is the midpoint inside BC, so $BM=CM$ ; together with the given $AM=BM$ , this gives $AM=BM=CM$ .
By the two isosceles triangles, let $\alpha=\angle BAM=\angle ABM$ and $\beta=\angle MAC=\angle ACM$ .
Since M lies between B and C in a nondegenerate triangle, $\angle A=\alpha+\beta$ , $\angle B=\alpha$ , and $\angle C=\beta$ .
The triangle angle sum gives $(\alpha+\beta)+\alpha+\beta=180^{\circ}$ , hence $2(\alpha+\beta)=180^{\circ}$ .
Therefore $\angle BAC=\alpha+\beta=90^{\circ}$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Parallelogram properties, Pythagoras and congruence.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

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

ABCD is a rectangle with AC=10 cm. Find BD and name the special rectangle property.

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

Reference answer

$BD=10$ cm.

The diagonals of a rectangle are equal.
BD=AC=10 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

ABCD is a rhombus with AB=5 cm. Find CD and explain why no diagonal calculation is needed.

A rhombus has ____ equal sides.
$AB=BC=CD=DA=\square$ cm.
Therefore CD=____ cm; the information needed is a side length, not a diagonal length.

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

Reference answer

$CD=5$ cm.

Every side of a rhombus is equal.
CD=AB=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

A parallelogram has one right angle. Which special type must it be?

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

Reference answer

A rectangle.

Use the one-right-angle criterion with the parallelogram premise.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

A parallelogram has AB=BC. Which special type follows?

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

Reference answer

A rhombus.

Equal adjacent sides plus equal opposite sides give all four sides equal.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

ABCD is a parallelogram and AC=BD. State the strongest classification justified by these conditions.

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

Reference answer

A rectangle; equal diagonals alone do not also guarantee a square.

Use equal diagonals with the known parallelogram.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

ABCD is a parallelogram and AC⊥BD. Which special type follows? Explain the required premise.

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

Reference answer

A rhombus. Perpendicular diagonals are used together with the parallelogram premise.

A general quadrilateral with perpendicular diagonals need not have equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

∠A=90° in △ABC, BC=10 cm, and M is the midpoint of BC. Find AM.

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

Reference answer

$AM=5$ cm.

The median to the hypotenuse of a right triangle equals half the hypotenuse.
AM=10/2=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A rhombus has diagonals AC=6 cm and BD=8 cm, meeting at O. Find its side AB, explaining both diagonal properties used.

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

Reference answer

AB=5 cm. The diagonals are perpendicular and bisect each other, giving legs 3 and 4.

AO=3 and BO=4, with ∠AOB=90°.
AB²=3²+4²=25, so AB=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Is every rectangle a parallelogram? Explain using the inclusive definition.

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

Reference answer

Yes. A rectangle is a special parallelogram.

Its opposite sides remain parallel even with the extra right-angle condition.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Must every rhombus be a rectangle? Use the shown non-square rhombus.

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

Reference answer

No. Equal sides do not force a right angle.

This rhombus has perpendicular diagonals but unequal diagonal lengths.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

A rectangle has AC=10 cm and diagonals meeting at O. Find BO.

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

Reference answer

$BO=5$ cm.

BD=AC=10 and O bisects BD.
BO=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

A general quadrilateral has equal diagonals. Must it be a rectangle? Explain with the shown trapezium.

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

Reference answer

No. The parallelogram premise is missing.

The equal-diagonal isosceles trapezium has only one pair of parallel opposite sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

A general quadrilateral has perpendicular diagonals. Must it be a rhombus? Give a reason from the shown kite.

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

Reference answer

No. The two upper sides and two lower sides have different lengths.

Perpendicularity alone does not guarantee diagonal bisection or four equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

In right △ABC, M is the midpoint of hypotenuse BC and AM=10 cm. Find BC.

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

Reference answer

$BC=20$ cm.

AM=BC/2, so BC=20 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

A rhombus has diagonals 6 cm and 8 cm. Derive its area by splitting it into four right triangles.

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

Reference answer

24 cm². Each triangle has legs 3 and 4, area 6, and there are four.

The diagonals are perpendicular and bisect each other.
Total area=4×(3×4/2)=24 cm².

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

In a nondegenerate triangle ABC, M is the midpoint of BC and AM=BM. Show that ∠BAC is a right angle, using angles in the two isosceles triangles.

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

Reference answer

AM=BM=CM. Let ∠B=x and ∠C=y. The angle at A is x+y, so 2(x+y)=180° and ∠A=90°.

M is the midpoint inside BC, so $BM=CM$ ; together with the given $AM=BM$ , this gives $AM=BM=CM$ .
By the two isosceles triangles, let $\alpha=\angle BAM=\angle ABM$ and $\beta=\angle MAC=\angle ACM$ .
Since M lies between B and C in a nondegenerate triangle, $\angle A=\alpha+\beta$ , $\angle B=\alpha$ , and $\angle C=\beta$ .
The triangle angle sum gives $(\alpha+\beta)+\alpha+\beta=180^{\circ}$ , hence $2(\alpha+\beta)=180^{\circ}$ .
Therefore $\angle BAC=\alpha+\beta=90^{\circ}$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

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

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22

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

Rectangles and rhombi: diagonal conditions and converses

Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

ABCD is a rectangle with AC=10 cm. Find BD and name the special rectangle property.

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

Guided attempt2

ABCD is a rhombus with AB=5 cm. Find CD and explain why no diagonal calculation is needed.

A rhombus has ____ equal sides.
$AB=BC=CD=DA=\square$ cm.
Therefore CD=____ cm; the information needed is a side length, not a diagonal length.

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

Prerequisite3

A parallelogram has one right angle. Which special type must it be?

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

Prerequisite4

A parallelogram has AB=BC. Which special type follows?

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

Rectangles and rhombi: diagonal conditions and converses

Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

ABCD is a parallelogram and AC=BD. State the strongest classification justified by these conditions.

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

Independent practice6

ABCD is a parallelogram and AC⊥BD. Which special type follows? Explain the required premise.

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

Independent practice7

∠A=90° in △ABC, BC=10 cm, and M is the midpoint of BC. Find AM.

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

Independent practice8

A rhombus has diagonals AC=6 cm and BD=8 cm, meeting at O. Find its side AB, explaining both diagonal properties used.

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

Exit question9

Is every rectangle a parallelogram? Explain using the inclusive definition.

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

Exit question10

Must every rhombus be a rectangle? Use the shown non-square rhombus.

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

Rectangles and rhombi: diagonal conditions and converses — Further practice

Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22

Further practice11

A rectangle has AC=10 cm and diagonals meeting at O. Find BO.

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

Further practice12

A general quadrilateral has equal diagonals. Must it be a rectangle? Explain with the shown trapezium.

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

Further practice13

A general quadrilateral has perpendicular diagonals. Must it be a rhombus? Give a reason from the shown kite.

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

Further practice14

In right △ABC, M is the midpoint of hypotenuse BC and AM=10 cm. Find BC.

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

Extension15

A rhombus has diagonals 6 cm and 8 cm. Derive its area by splitting it into four right triangles.

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

Extension16

In a nondegenerate triangle ABC, M is the midpoint of BC and AM=BM. Show that ∠BAC is a right angle, using angles in the two isosceles triangles.

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

Rectangles and rhombi: diagonal conditions and converses — Teacher answers

Year 8, lower volume · text pp. 37–44 · textbook lessons 19–22

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Parallelogram properties, Pythagoras and congruence.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict which diagonal properties a non-square rectangle and a non-square rhombus share.
8′ On paper compare a 4-by-3 rectangle (diagonals 5) with the rhombus (0,0),(3,4),(6,0),(3,−4) (diagonals 6 and 8). Record bisection, equal lengths and perpendicularity separately. Use the stated shapes as premises, not their apparent outlines.
5′ Identify the shared bisection property and the differing equal/perpendicular properties. Keep the parallelogram premise when using diagonal criteria.
2′ Individually judge whether perpendicular diagonals alone force an arbitrary quadrilateral to be a rhombus; give a missing condition.

Worked example1

ABCD is a rectangle with AC=10 cm. Find BD and name the special rectangle property.

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

Reference answer

$BD=10$ cm.

The diagonals of a rectangle are equal.
BD=AC=10 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

ABCD is a rhombus with AB=5 cm. Find CD and explain why no diagonal calculation is needed.

A rhombus has ____ equal sides.
$AB=BC=CD=DA=\square$ cm.
Therefore CD=____ cm; the information needed is a side length, not a diagonal length.

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

Reference answer

$CD=5$ cm.

Every side of a rhombus is equal.
CD=AB=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

A parallelogram has one right angle. Which special type must it be?

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

Reference answer

A rectangle.

Use the one-right-angle criterion with the parallelogram premise.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

A parallelogram has AB=BC. Which special type follows?

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

Reference answer

A rhombus.

Equal adjacent sides plus equal opposite sides give all four sides equal.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

ABCD is a parallelogram and AC=BD. State the strongest classification justified by these conditions.

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

Reference answer

A rectangle; equal diagonals alone do not also guarantee a square.

Use equal diagonals with the known parallelogram.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

ABCD is a parallelogram and AC⊥BD. Which special type follows? Explain the required premise.

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

Reference answer

A rhombus. Perpendicular diagonals are used together with the parallelogram premise.

A general quadrilateral with perpendicular diagonals need not have equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

∠A=90° in △ABC, BC=10 cm, and M is the midpoint of BC. Find AM.

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

Reference answer

$AM=5$ cm.

The median to the hypotenuse of a right triangle equals half the hypotenuse.
AM=10/2=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A rhombus has diagonals AC=6 cm and BD=8 cm, meeting at O. Find its side AB, explaining both diagonal properties used.

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

Reference answer

AB=5 cm. The diagonals are perpendicular and bisect each other, giving legs 3 and 4.

AO=3 and BO=4, with ∠AOB=90°.
AB²=3²+4²=25, so AB=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Is every rectangle a parallelogram? Explain using the inclusive definition.

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

Reference answer

Yes. A rectangle is a special parallelogram.

Its opposite sides remain parallel even with the extra right-angle condition.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Must every rhombus be a rectangle? Use the shown non-square rhombus.

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

Reference answer

No. Equal sides do not force a right angle.

This rhombus has perpendicular diagonals but unequal diagonal lengths.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

A rectangle has AC=10 cm and diagonals meeting at O. Find BO.

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

Reference answer

$BO=5$ cm.

BD=AC=10 and O bisects BD.
BO=5 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

A general quadrilateral has equal diagonals. Must it be a rectangle? Explain with the shown trapezium.

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

Reference answer

No. The parallelogram premise is missing.

The equal-diagonal isosceles trapezium has only one pair of parallel opposite sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

A general quadrilateral has perpendicular diagonals. Must it be a rhombus? Give a reason from the shown kite.

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

Reference answer

No. The two upper sides and two lower sides have different lengths.

Perpendicularity alone does not guarantee diagonal bisection or four equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

In right △ABC, M is the midpoint of hypotenuse BC and AM=10 cm. Find BC.

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

Reference answer

$BC=20$ cm.

AM=BC/2, so BC=20 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

A rhombus has diagonals 6 cm and 8 cm. Derive its area by splitting it into four right triangles.

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

Reference answer

24 cm². Each triangle has legs 3 and 4, area 6, and there are four.

The diagonals are perpendicular and bisect each other.
Total area=4×(3×4/2)=24 cm².

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

In a nondegenerate triangle ABC, M is the midpoint of BC and AM=BM. Show that ∠BAC is a right angle, using angles in the two isosceles triangles.

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

Reference answer

AM=BM=CM. Let ∠B=x and ∠C=y. The angle at A is x+y, so 2(x+y)=180° and ∠A=90°.

M is the midpoint inside BC, so $BM=CM$ ; together with the given $AM=BM$ , this gives $AM=BM=CM$ .
By the two isosceles triangles, let $\alpha=\angle BAM=\angle ABM$ and $\beta=\angle MAC=\angle ACM$ .
Since M lies between B and C in a nondegenerate triangle, $\angle A=\alpha+\beta$ , $\angle B=\alpha$ , and $\angle C=\beta$ .
The triangle angle sum gives $(\alpha+\beta)+\alpha+\beta=180^{\circ}$ , hence $2(\alpha+\beta)=180^{\circ}$ .
Therefore $\angle BAC=\alpha+\beta=90^{\circ}$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.