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Level 7-8 Shapes - Pythagoras

Right triangles unlock distances. Use Pythagoras’ theorem to find missing sides, check for right angles, and solve real problems involving ladders, ramps, and maps.

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Fascinating Fact:

If a square tile has a side of 8 cm, the diagonal across it measures √(8² + 8²) = √128 ≈ 11.3 cm.

In KS3 Maths, Pythagoras’ theorem links the sides of a right-angled triangle. You’ll calculate missing lengths, test whether triangles are right-angled, and apply the theorem to maps, ramps and real measurements.

Key Terms

Right-angled triangle: A triangle with one 90° angle.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Pythagoras’ theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.

Frequently Asked Questions (Click to see answers)

What is Pythagoras’ theorem in KS3?

Pythagoras’ theorem says that in any right-angled triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse: a² + b² = c².

How do I find a missing side using Pythagoras?

If the legs are 5 and 12, the hypotenuse is √(5² + 12²) = √169 = 13. If the hypotenuse and one leg are known, rearrange to a = √(c² ? b²).

How can I check if a triangle is right-angled?

Use the converse: if the longest side squared equals the sum of the other two squares (e.g., 7² + 24² = 25²), then the triangle is right-angled.

Try These Related Quizzes

1 .

The foot of a ladder of length 3.75 m is 1.4 m away from a vertical wall. How high up the wall will the ladder reach (to the nearest 1 cm)?

3.84 m

3.48 m

3.24 m

3.18 m

This assumes that the ground is perfectly level!

2 .

The hypotenuse of a right-angled triangle is 26 cm. One of the shorter sides is 10 cm. How long is the third side?

30 cm

24 cm

16 cm

8 cm

26² - 10² = 24²

3 .

If the longest side of a right-angled triangle is c cm and the other sides are a cm and b cm, Pythagoras' Theorem states that .......

a² x b² = c²

(a + b)² = c²

a² + b² = c²

a² - b² = c²

Remember, Pythagoras' Theorem only applies to right-angled triangles

4 .

In a triangle, base 25 cm and height 12 cm, the perpendicular from the base to the opposite vertex divides the base in the ratio 3:2. How long are the other two sides (to 1dp)?

15.6 cm and 19.2 cm

16.3 cm and 18.9 cm

15.6 cm and 16.3 cm

18.9 cm and 19.2 cm

Split the triangle into two right-angled triangles with bases 15 cm and 10 cm

5 .

A right-angled isosceles triangle has hypotenuse of length 15 cm. What are the angles at either end of the hypotenuse?

15^o and 75^o

30^o and 60^o

45^o and 45^o

20^o and 70^o

The side length is irrelevant. Any right-angled isosceles triangle has two angles of 45^o

6 .

The following sets of numbers represent the lengths of the sides of a triangle. Which is not a right-angled triangle?

15, 20, 25

18, 24, 30

16, 30, 36

15, 36, 39

Trial and error is a very useful tool in maths

7 .

The cross-section of a porch roof is an isosceles triangle with height 0.8 m and base 3.0 m. How long is each sloping roof section?

3.1 m

2.3 m

1.9 m

1.7 m

I hope you remembered the rules for isosceles triangles

8 .

A rectangle has length 2.4 m and width 0.7 m. How long is its diagonal?

2.0 m

2.5 m

3.0 m

3.5 m

The diagonal is the hypotenuse of a right-angled triangle

9 .

In a right-angled triangle the sides forming the 90^o angle are 6 cm and 8 cm. What is the length of the third side?

10 cm

12 cm

14 cm

16 cm

36 + 64 = 100; ?100 = 10. The longest side in a right-angled triangle is called the hypotenuse

10 .

A triangle has sides of length 4 cm, 5 cm and 6 cm. Which of these statements is not true?

The three angles add up to 180^o

All three angles are acute

The triangle is scalene

The largest angle is 90^o

(4 x 4) + (5 x 5) = 41 but 6 x 6 = 36 so the triangle cannot contain a right angle

You can find more about this topic by visiting BBC Bitesize - Pythagoras

Author: Frank Evans (Specialist 11 Plus Teacher and Tutor)

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