PAPER · REASONING · PRACTICE

Pythagoras: identify the sides and calculate lengths

Model right triangles, choose addition or subtraction of squares, and retain positive exact lengths.

40 minutes · Year 8, lower volume · text pp. 13–18 · textbook lessons 7–9

Textbook reference: Year 8, lower volume · Chapter 17: Pythagorean theorem · lessons 7–9

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.

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

3 min: Predict how the hypotenuse and its square change when both legs double. Name the hypotenuse first. 8 min: In the existing Pythagoras inquiry, record legs (3,4) then (6,8): c=5,10 and c²=25,100. Next change only the first leg from 3 to 6 while the second stays 4; keep exact square relations separate from decimal readouts. 5 min: Explain the length factor 2 and square factor 4 using the theorem. Why does changing only one leg fail to produce the same scale factor? 2 min: Individually find the hypotenuse for legs 9 and 12, stating the right-angle condition and units.

Organise the key idea

For a right triangle with hypotenuse c, $a^{2}+b^{2}=c^{2}$ . A length is the positive square root.

Open the original chapter Open the interactive inquiry

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

A right triangle has legs $5$ cm and $12$ cm. Find its hypotenuse.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $5^{2}+12^{2}$ ; take the positive square root.

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

A right triangle has hypotenuse $10$ cm and one leg $6$ cm. Find the other leg.

Let the missing leg be b: $b^{2}+6^{2}=10^{2}$ .
$b^{2}=100-36=\square$ .
Positive length: $b=\sqrt{\square}=\square$ cm.

The 10 cm side is the hypotenuse, so its square is the total of the two leg squares.
Isolate the missing square: $b^{2}=10^{2}-6^{2}$ .
The difference of squares is 64. Take its positive square root because b is a length.

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $10^{2}-6^{2}$ ; take the positive square root.

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

Calculate $3^{2}+4^{2}$ .

3^{2}+4^{2}

Do not add the sides before squaring.
The two squares are 9 and 16.
Add the two squares.

Square before adding: 9+16=25.

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

In the diagram, ∠C is a right angle. Which side is the hypotenuse?

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

Use the right-angle mark.
The hypotenuse lies opposite that angle.
Its endpoints are A and B.

Locate C, then choose the side not touching C.

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

A right triangle has legs $7$ cm and $24$ cm. Find its hypotenuse.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $7^{2}+24^{2}$ ; take the positive square root.

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

A right triangle has hypotenuse $17$ cm and one leg $8$ cm. Find the other leg.

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $17^{2}-8^{2}$ ; take the positive square root.

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 rectangle has adjacent sides 6 cm and 8 cm. Find its diagonal, naming the right triangle used.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $6^{2}+8^{2}$ ; take the positive square root.

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 13 m ladder leans against a vertical wall. Its foot is 5 m from the wall on level ground. Find the height reached.

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $13^{2}-5^{2}$ ; take the positive square root.

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

A right triangle has legs $8$ cm and $15$ cm. Find its hypotenuse.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $8^{2}+15^{2}$ ; take the positive square root.

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

A right triangle has hypotenuse $25$ cm and one leg $7$ cm. Find the other leg.

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $25^{2}-7^{2}$ ; take the positive square root.

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 right triangle has legs $1.5$ cm and $2$ cm. Find its hypotenuse.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $1.5^{2}+2^{2}$ ; take the positive square root.

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 square has side 5 cm. Find its diagonal exactly.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $5^{2}+5^{2}$ ; take the positive square root.

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

Find the distance from $(0,0)$ to $(3,4)$ by completing a right triangle.

Use the stated conditions and the labelled scales.

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $3^{2}+4^{2}$ ; take the positive square root.

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

A right triangle has hypotenuse $29$ cm and one leg $20$ cm. Find the other leg.

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $29^{2}-20^{2}$ ; take the positive square root.

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 right triangle has legs 4 cm and 5 cm. Find the exact hypotenuse and explain why 9 cm is impossible.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

The square is $4^{2}+5^{2}=41$ , so the length is $\sqrt{41}$ cm.
The sum 9 is not the square-root calculation; equality with the two-leg sum would be degenerate.

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

A right triangle with legs 3 cm and 4 cm is enlarged by scale factor 2. Find its new hypotenuse and explain why it doubles.

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

Locate the right angle and the opposite hypotenuse.
Write the square relation before substituting.
A length is positive; retain the stated unit.

New legs are 6 cm and 8 cm. Their squares total 100, so the hypotenuse is 10 cm.
Scaling each leg by 2 multiplies the sum of squares by 4; its positive square root scales by 2.

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

Squares, roots, perpendicular lines and units of length.

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.

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. 13–18 · textbook lessons 7–9

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

Pythagoras: identify the sides and calculate lengths

Year 8, lower volume · text pp. 13–18 · textbook lessons 7–9 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A right triangle has legs $5$ cm and $12$ cm. Find its hypotenuse.

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

Guided attempt2

A right triangle has hypotenuse $10$ cm and one leg $6$ cm. Find the other leg.

Let the missing leg be b: $b^{2}+6^{2}=10^{2}$ .
$b^{2}=100-36=\square$ .
Positive length: $b=\sqrt{\square}=\square$ cm.

Prerequisite3

Calculate $3^{2}+4^{2}$ .

3^{2}+4^{2}

Prerequisite4

In the diagram, ∠C is a right angle. Which side is the hypotenuse?

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

Pythagoras: identify the sides and calculate lengths

Year 8, lower volume · text pp. 13–18 · textbook lessons 7–9 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

A right triangle has legs $7$ cm and $24$ cm. Find its hypotenuse.

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

Independent practice6

A right triangle has hypotenuse $17$ cm and one leg $8$ cm. Find the other leg.

Independent practice7

A rectangle has adjacent sides 6 cm and 8 cm. Find its diagonal, naming the right triangle used.

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

Independent practice8

A 13 m ladder leans against a vertical wall. Its foot is 5 m from the wall on level ground. Find the height reached.

Exit question9

A right triangle has legs $8$ cm and $15$ cm. Find its hypotenuse.

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

Exit question10

A right triangle has hypotenuse $25$ cm and one leg $7$ cm. Find the other leg.

Pythagoras: identify the sides and calculate lengths — Further practice

Year 8, lower volume · text pp. 13–18 · textbook lessons 7–9

Further practice11

A right triangle has legs $1.5$ cm and $2$ cm. Find its hypotenuse.

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

Further practice12

A square has side 5 cm. Find its diagonal exactly.

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

Further practice13

Find the distance from $(0,0)$ to $(3,4)$ by completing a right triangle.

Use the stated conditions and the labelled scales.

Further practice14

A right triangle has hypotenuse $29$ cm and one leg $20$ cm. Find the other leg.

Extension15

A right triangle has legs 4 cm and 5 cm. Find the exact hypotenuse and explain why 9 cm is impossible.

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

Extension16

A right triangle with legs 3 cm and 4 cm is enlarged by scale factor 2. Find its new hypotenuse and explain why it doubles.

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

Pythagoras: identify the sides and calculate lengths — Teacher answers

Year 8, lower volume · text pp. 13–18 · textbook lessons 7–9

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: Squares, roots, perpendicular lines and units of length.

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

3′ Predict how the hypotenuse and its square change when both legs double. Name the hypotenuse first.
8′ In the existing Pythagoras inquiry, record legs (3,4) then (6,8): c=5,10 and c²=25,100. Next change only the first leg from 3 to 6 while the second stays 4; keep exact square relations separate from decimal readouts.
5′ Explain the length factor 2 and square factor 4 using the theorem. Why does changing only one leg fail to produce the same scale factor?
2′ Individually find the hypotenuse for legs 9 and 12, stating the right-angle condition and units.

Worked example1

A right triangle has legs $5$ cm and $12$ cm. Find its hypotenuse.

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

Reference answer

$13$ cm.

13

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $5^{2}+12^{2}$ ; take the positive square root.

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

Guided attempt2

A right triangle has hypotenuse $10$ cm and one leg $6$ cm. Find the other leg.

Let the missing leg be b: $b^{2}+6^{2}=10^{2}$ .
$b^{2}=100-36=\square$ .
Positive length: $b=\sqrt{\square}=\square$ cm.

Reference answer

$8$ cm.

8

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $10^{2}-6^{2}$ ; take the positive square root.

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

Prerequisite3

Calculate $3^{2}+4^{2}$ .

3^{2}+4^{2}

Reference answer

$25$

25

Square before adding: 9+16=25.

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

Prerequisite4

In the diagram, ∠C is a right angle. Which side is the hypotenuse?

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

Reference answer

AB, the side opposite the right angle.

Locate C, then choose the side not touching C.

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

Independent practice5

A right triangle has legs $7$ cm and $24$ cm. Find its hypotenuse.

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

Reference answer

$25$ cm.

25

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $7^{2}+24^{2}$ ; take the positive square root.

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

Independent practice6

A right triangle has hypotenuse $17$ cm and one leg $8$ cm. Find the other leg.

Reference answer

$15$ cm.

15

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $17^{2}-8^{2}$ ; take the positive square root.

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 rectangle has adjacent sides 6 cm and 8 cm. Find its diagonal, naming the right triangle used.

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

Reference answer

$10$ cm.

10

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $6^{2}+8^{2}$ ; take the positive square root.

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 13 m ladder leans against a vertical wall. Its foot is 5 m from the wall on level ground. Find the height reached.

Reference answer

$12$ m.

12

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $13^{2}-5^{2}$ ; take the positive square root.

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

Exit question9

A right triangle has legs $8$ cm and $15$ cm. Find its hypotenuse.

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

Reference answer

$17$ cm.

17

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $8^{2}+15^{2}$ ; take the positive square root.

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

Exit question10

A right triangle has hypotenuse $25$ cm and one leg $7$ cm. Find the other leg.

Reference answer

$24$ cm.

24

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $25^{2}-7^{2}$ ; take the positive square root.

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 right triangle has legs $1.5$ cm and $2$ cm. Find its hypotenuse.

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

Reference answer

$2.5$ cm.

2.5

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $1.5^{2}+2^{2}$ ; take the positive square root.

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 square has side 5 cm. Find its diagonal exactly.

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

Reference answer

$5\sqrt{2}$ cm.

5\sqrt{2}

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $5^{2}+5^{2}$ ; take the positive square root.

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

Further practice13

Find the distance from $(0,0)$ to $(3,4)$ by completing a right triangle.

Use the stated conditions and the labelled scales.

Reference answer

$5$ units.

5

The right angle is given; add the two leg squares to find the hypotenuse square.
The required square is $3^{2}+4^{2}$ ; take the positive square root.

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

Use the stated conditions and the labelled scales.

Further practice14

A right triangle has hypotenuse $29$ cm and one leg $20$ cm. Find the other leg.

Reference answer

$21$ cm.

21

The hypotenuse is known; subtract the square of the known leg from its square.
The required square is $29^{2}-20^{2}$ ; take the positive square root.

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

Extension15

A right triangle has legs 4 cm and 5 cm. Find the exact hypotenuse and explain why 9 cm is impossible.

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

Reference answer

$\sqrt{41}$ cm.

\sqrt{41}

The square is $4^{2}+5^{2}=41$ , so the length is $\sqrt{41}$ cm.
The sum 9 is not the square-root calculation; equality with the two-leg sum would be degenerate.

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

Extension16

A right triangle with legs 3 cm and 4 cm is enlarged by scale factor 2. Find its new hypotenuse and explain why it doubles.

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

Reference answer

$10$ cm.

10

New legs are 6 cm and 8 cm. Their squares total 100, so the hypotenuse is 10 cm.
Scaling each leg by 2 multiplies the sum of squares by 4; its positive square root scales by 2.

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