Using Pythagoras' theorem to justify a right-angled triangle | Maths

Starter quiz

Pythagoras’ theorem states that the sum of the squares of the two shorter sides of a __________ triangle is equal to the square of the hypotenuse.

scalene

isosceles

equilateral

right-angled ✓

obtuse-angled
Put the stages of working in order.

1

⇔

$x^2=8^2+15^2$

2

⇔

$x^2=64+225$

3

⇔

$x^2=289$

4

⇔

$x=\sqrt{289}$

5

⇔

$x=17$

6

⇔

The hypotenuse is 17 cm.
Work out the length of the hypotenuse.

'26 cm' ✓
Work out the length of $x$ .

'6 cm' ✓
Select the correct answers for $x$ .

$\sqrt{87}$ cm ✓

9.3 cm (rounded to the nearest tenth) ✓

9.32 cm (rounded to 3 significant figures)

9.32 cm (truncated to 2 decimal places) ✓

9.32738 (rounded to 5 significant figures)
A trapezium is formed by a right-angled triangle and a rectangle. Work out the length of the edge marked $x$ .

'29 cm' ✓

If a triangle is right-angled this means...

...one of the angles is $90^\circ$ . ✓

...the longest edge is called the hypotenuse. ✓

…there are three different angles.

...Pythagoras' theorem holds. ✓

...the other two angles are $45^\circ$ .
If a triangle is right-angled then $a^2+b^2=c^2$ where...

... $a$ is the hypotenuse.

... $b$ is the hypotenuse.

... $c$ is the hypotenuse. ✓
A triangle has these three edge lengths: 13 cm, 18cm and 31 cm. Is it a right-angled triangle?

Yes, as 13 + 18 = 31.

No, as $13^2+18^2\neq 31^2$ . ✓

Impossible to tell, as we haven't got a diagram.
An isosceles triangle has these three edge lengths: $19\sqrt{2}$ cm, $19\sqrt{2}$ cm and 38 cm. Is it a right-angled triangle?

No, you cannot have a right-angled isosceles triangle.

Yes, as Pythagoras' theorem holds. ✓

Impossible to tell, as isosceles triangles can have many different angles.
Here is a plan view of a room. There are two external walls; are they perpendicular?

Yes - Pythagoras' theorem holds.

No - Pythagoras' theorem does not hold. ✓

There is not enough information.
Which of these triangles ABC, are right-angled?

AB = 7 cm, AC = 9 cm, BC = 16 cm

AB = 6 cm, AC = 8 cm, BC = 10 cm ✓

AB = 8 cm, AC = 14 cm, BC = 22 cm

AB = 20 cm, AC = 99 cm, BC = 101 cm ✓

AB = 40 cm, AC = 198 cm, BC = 202 cm ✓

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Incorrectly rearranging Pythagoras' theorem.

It can be helpful to scaffold the steps for any pupils who are struggling with rearranging.

Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.
Pythagoras' theorem - Pythagoras’ theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.