KS3 Maths · Year 8 · Chapter 6: Pythagoras' theorem · Lesson 1

Introducing Pythagoras' theorem

Year 8About 70 minutesCalculator allowed45 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $6^{2} + 1 1^{2}$ .
[1 mark]
Answer
A2
Needed today
Write down the value of $196$ .
[1 mark]
Answer
A3
Last lesson
A number doubles every year. It is $40$ now. What is it after $3$ years?
[1 mark]
Answer
A4
Year 7
A triangle has base $10$ cm and height $7$ cm. Work out its area.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it. Give lengths to $1$ decimal place unless they are exact.

Example 1

Work out the length of the hypotenuse, $x$ .

Hint The hypotenuse is opposite the right angle. Square the two shorter sides, add, then square root.

B1
Work out the length of the hypotenuse, $x$ . Give your answer to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots$ cm

Example 2

Work out the length of the side marked $x$ . Give your answer to $1$ decimal place.

Hint

x

is a shorter side:

11

cm is opposite the right angle. Subtract the squares, then square root.

B2
Work out the length of the side marked $y$ . Give your answer to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $y = \dots$ cm

Example 3

The triangle is isosceles. Work out (a) its height, $h$ , (b) its area. Give your answers to $1$ decimal place.

Hint The height cuts the base in half: a right-angled triangle with hypotenuse

9

cm and side

4

cm.

B3
The triangle is isosceles. Work out (a) its height, $h$ , (b) its area. Give your answers to $1$ decimal place.
[4 marks]
Not drawn accurately

Example 4

In triangle $A B C$ , $A D$ is perpendicular to $B C$ . Work out the length of $A C$ . Give your answer to $1$ decimal place.

Hint First use triangle

A B D

to find

A D

, the side both triangles share.

B4
In triangle $A B C$ , $A D$ is perpendicular to $B C$ . Work out the length of $A C$ .
[3 marks]
Not drawn accurately
Answer $A C = \dots$ cm

Practice

Year 8

C1
Triangle $P QR$ has a right angle at $Q$ . Which side is the hypotenuse? Write Pythagoras' theorem for this triangle, using the letters.
[2 marks]
C2
A rectangle is $16$ cm long and $70$ mm wide. Work out the length of its diagonal, in cm.
[3 marks]
Not drawn accurately
Answer $\dots$ cm

C3
Work out the distance between the points $A$ and $B$ . Each square is $1$ unit.
[3 marks]
Answer $\dots$ units
C4
Squares are drawn on the two shorter sides of a right-angled triangle. Their areas are $30$ cm² and $45$ cm². Work out the length of the hypotenuse.
[2 marks]
Not drawn accurately
Answer $\dots$ cm

Exam-style questions

GCSE Foundation

D1
A right-angled triangle has a hypotenuse of $12$ cm and one shorter side of $7$ cm. Lily works out the other side like this: $1 2^{2} + 7^{2} = 193$ , $193 = 13.9$ cm.
Explain what Lily has done wrong. Work out the correct length, to $1$ decimal place.
[3 marks]

D2
$A B C D$ is a quadrilateral. Angle $A B C = 9 0^{\circ}$ and angle $A C D = 9 0^{\circ}$ .
Not drawn accurately
(a)
Show that $A C^{2} = 74$ .
[1 mark]
(b)
Work out the length of $A D$ . Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots$ cm
(c)
Work out the perimeter of $A B C D$ . Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots$ cm
Total for D2: 5 marks
D3
The diagram shows an isosceles triangle. Its base is $14$ cm and its area is $84$ cm². Work out the length, $x$ , of each sloping side. Give your answer to $1$ decimal place.
[4 marks]
Not drawn accurately
Answer $x = \dots$ cm
D4
$P$ is the point $(- 3, - 1)$ , $Q$ is the point $(5, 3)$ and $R$ is the point $(4, - 6)$ .
(a)
Work out the length of $P Q$ . Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots$ units
(b)
Which point, $Q$ or $R$ , is further from $P$ ? You must show your working.
[2 marks]
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A rectangle has a diagonal of $25$ cm and a perimeter of $62$ cm. Work out the area of the rectangle.
[4 marks]
Hint 1 · What to try
Call the sides $a$ and $b$ . What do you know about $a + b$ , and about $a^{2} + b^{2}$ ?
Hint 2 · The first line
$a + b = 31$ and $a^{2} + b^{2} = 625$ . Try pairs of whole numbers that add to $31$ .
Hint 3 · The full method
$7 + 24 = 31$ and $7^{2} + 2 4^{2} = 49 + 576 = 625$ . So the sides are $7$ cm and $24$ cm, and the area is $7 \times 24 = 168$ cm².
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the hypotenuse of a right-angled triangle.
I can find a shorter side of a right-angled triangle.
I can use Pythagoras' theorem for diagonals, isosceles triangles and points on a grid.
I can split a shape into right-angled triangles to find a length.

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