Pythagoras

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Notes de leçon

Pythagoras' Theorem

Pythagoras was a Greek mathematician who lived over 2500 years ago.
Pythagoras' theorem links the lengths of the three sides of a right-angled triangle.
The hypotenuse is the longest side, opposite the right angle.
Formula: a² + b² = c², where c is the hypotenuse and a, b are the shorter sides.
It does not matter which shorter side is labelled a or b.

Right-angled triangle

Finding the Hypotenuse

To find the hypotenuse: square the two shorter sides, add them, then take the positive square root.
Formula: c = √(a² + b²).
When finding the hypotenuse, you add inside the square root.
Example: If a = 3, b = 4, then c = √(3² + 4²) = √25 = 5.

Finding a Shorter Side

To find a shorter side: square the hypotenuse and the other shorter side, subtract the smaller from the larger, then take the square root.
Formula: a = √(c² – b²).
When finding a shorter side, you subtract inside the square root.
Ensure the hypotenuse is longer than any other side; otherwise, check your working.

Using Pythagoras with Other Shapes

Pythagoras can be applied to any shape that can be split into right-angled triangles.
For a rectangle or square, the diagonal splits the shape into two identical right-angled triangles.
The diagonal becomes the hypotenuse of those triangles.
Example: In a rectangle 14 cm by w cm with diagonal 23 cm, use Pythagoras to find w: w = √(23² – 14²).

Multi-Step Problems

In multi-step problems, leave intermediate answers as exact values (e.g., √63) to avoid rounding errors.
Only round the final answer to the required degree of accuracy.
Work through each right-angled triangle step by step.
Example: Find BC in a trapezium by first finding BD using triangle ABD, then DC = AC – AD, then BC = √(BD² + DC²).

Worked example: two right-angled triangles

Common Mistakes & Tips

The hypotenuse must be the longest side; if it isn't, you've made an error.
When subtracting for a shorter side, always subtract the smaller square from the larger square.
Otherwise, you'll get a negative number and a 'Math Error' when taking the square root.
Always check that your answer is reasonable.

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1.Which side of a right-angled triangle is the hypotenuse?
Easy
- AThe side opposite the right angle
- BThe side adjacent to the right angle
- CThe shortest side
- DAny side
2.Pythagoras' theorem states that for a right-angled triangle with sides a, b and hypotenuse c:
Easy
- Aa² + b² = c²
- Ba + b = c
- Ca² = b² + c²
- Da² - b² = c²
3.A right-angled triangle has legs of lengths 3 cm and 4 cm. What is the length of the hypotenuse?
Easy
- A5 cm
- B7 cm
- C12 cm
- D25 cm
4.A flagpole is 25 m tall. A rope from the top is tied to the ground 8 m from the base. What is the length of the rope?
Medium
- A26.2 m
- B24.0 m
- C33.0 m
- D17.0 m
5.A rectangle is 14 cm wide and has a diagonal of 23 cm. What is the length of the rectangle?
Medium
- A18.2 cm
- B26.9 cm
- C9.0 cm
- D18.0 cm
6.A right-angled triangle has legs of lengths x and 8 cm, and hypotenuse 17 cm. What is x?
Easy
- A15 cm
- B18.8 cm
- C9 cm
- D25 cm
7.The length of one side of a rectangle is 12 cm and the diagonal is 13 cm. What is the area of the rectangle?
Medium
- A60 cm²
- B156 cm²
- C30 cm²
- D144 cm²
8.A trapezium has AB = 150 m, BC = 90 m, CD = 120 m, with angle ABC = angle BCD = 90°. What is the length AD, correct to the nearest metre?
Hard
- A95 m
- B85 m
- C108 m
- D122 m

Plus de questions

9.A rectangular pond ABCD has AB = 12 m and diagonal AC = 15 m. What is the length BC?
Easy
- A9 m
- B3 m
- C19.2 m
- D5 m
10.A right-angled triangle has sides 5 cm, 12 cm and 13 cm. What is its perimeter?
Easy
- A30 cm
- B25 cm
- C17 cm
- D60 cm
11.A vertical flagpole is on horizontal ground. A guy rope of length 6.9 m is attached to the top and fixed to the ground 3.2 m from the base. What is the height of the flagpole?
Medium
- A6.1 m
- B7.6 m
- C3.7 m
- D5.5 m
12.A rectangular piece of wood ABCD has AB = 12 cm and AC = 15 cm. What is BC?
Easy
- A9 cm
- B3 cm
- C19.2 cm
- D5 cm
13.In a right-angled triangle, the side opposite the right angle is called the
Easy
- Ahypotenuse
- Badjacent
- Copposite
- Dbase
14.A right-angled triangle has legs 5 cm and 12 cm. What is the length of the hypotenuse?
Easy
- A13 cm
- B17 cm
- C7 cm
- D11 cm
15.A right-angled triangle has hypotenuse 10 cm and one leg 6 cm. What is the length of the other leg?
Easy
- A8 cm
- B4 cm
- C7 cm
- D9 cm
16.A right-angled triangle has legs of lengths x cm and 15 cm, and hypotenuse 25 cm. What is x?
Easy
- A20 cm
- B10 cm
- C30 cm
- D5 cm
17.The length of one side of a rectangle is 9 cm and the diagonal is 15 cm. What is the area of the rectangle?
Medium
- A108 cm²
- B135 cm²
- C72 cm²
- D96 cm²
18.A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?
Easy
- A4 m
- B2 m
- C6 m
- D3.5 m
19.In the diagram, triangle ABC is right-angled at B, with AB = 7 cm and BC = 24 cm. D is a point on AC such that BD ⟂ AC. Calculate the length BD.
Hard
- A6.72 cm
- B5.6 cm
- C7.2 cm
- D8.4 cm
20.A right-angled triangle has legs 9 cm and 12 cm. What is the length of the hypotenuse?
Easy
- A15 cm
- B21 cm
- C3 cm
- D225 cm
21.A rectangle has width 8 cm and diagonal 17 cm. What is the length of the rectangle?
Medium
- A15 cm
- B9 cm
- C25 cm
- D18.8 cm
22.In a right-angled triangle, the hypotenuse is 41.8 (correct to 3 s.f.) and one leg is 40. What is the other leg?
Medium
- A12
- B18
- C9
- D15
23.Which of the following is the correct formula to find the hypotenuse c given legs a and b?
Easy
- Ac = √(a² + b²)
- Bc = a² + b²
- Cc = √(a² - b²)
- Dc = a + b
24.In a right-angled triangle, the hypotenuse is always the side
Easy
- Aopposite the right angle
- Badjacent to the right angle
- Cshortest
- Dnext to the right angle
25.A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. What is the length of the other leg?
Medium
- A12 cm
- B8 cm
- C18 cm
- D√(194) cm
26.A rectangle has width 7 cm and diagonal 25 cm. What is the length of the rectangle?
Medium
- A24 cm
- B18 cm
- C32 cm
- D√(674) cm
27.A right-angled triangle has legs x and 9 cm, and hypotenuse 15 cm. What is x?
Medium
- A12 cm
- B6 cm
- C√(306) cm
- D24 cm
28.A rectangle has length 15 cm and diagonal 17 cm. What is its width?
Medium
- A8 cm
- B2 cm
- C√(514) cm
- D32 cm
29.A ladder 10 m long leans against a wall. The foot is 6 m from the wall. How high up the wall does the ladder reach?
Medium
- A8 m
- B4 m
- C√(136) m
- D16 m
30.In a right-angled triangle, which side is always the longest?
Easy
- AHypotenuse
- BAdjacent side
- COpposite side
- DAny side
31.Pythagoras' theorem states that for a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Easy
True or false?
32.A rectangle is 9 cm wide and has a diagonal of 15 cm. What is the length of the rectangle?
Medium
- A12 cm
- B10 cm
- C14 cm
- D18 cm
33.Match each formula to its correct description.
Medium
- c = √(a² + b²)
- a = √(c² - b²)
- b = √(c² - a²)
- Finding the hypotenuse
- Finding a shorter side
- Finding the other shorter side
34.Order the steps to find the length of the hypotenuse of a right-angled triangle with legs 5 cm and 12 cm.
Medium
- Take the square root of 169
- Add 25 and 144
- Square 5 and 12
- Write the answer: 13 cm
35.Which of the following statements about Pythagoras' theorem are true? (Select all that apply)
Medium
- AIt only works for right-angled triangles.
- BThe hypotenuse is always the longest side.
- CThe formula is a² + b² = c² where c is the shortest side.
- DYou can use it to find the diagonal of a rectangle.
36.In a right-angled triangle, if you know the lengths of the two shorter sides, you can find the length of the hypotenuse by squaring each shorter side, adding the squares, and then taking the square root.
Easy
True or false?
37.A right-angled triangle has legs of lengths 7 cm and 24 cm. What is the length of the hypotenuse?
Easy
- A25 cm
- B31 cm
- C23 cm
- D17 cm
38.A rectangle has a width of 9 cm and a diagonal of 41 cm. What is the length of the rectangle?
Medium
- A40 cm
- B42 cm
- C32 cm
- D50 cm
39.A right-angled triangle has legs of lengths 6 cm and 8 cm. What is the length of the hypotenuse?
Easy
40.A right-angled triangle has legs of lengths x cm and 24 cm, and hypotenuse 25 cm. What is x?
Medium
41.What is the name given to the longest side of a right-angled triangle?
Easy
42.Complete the statement of Pythagoras' theorem.
Easy
For a right-angled triangle with legs a and b and hypotenuse c, ____ + ____ = ____.
43.A right-angled triangle has hypotenuse 26 cm and one leg 10 cm. Find the length of the other leg.
Medium
44.Complete the sentence.
Easy
Pythagoras' theorem states that for a right-angled triangle, the square of the ____ is equal to the sum of the squares of the other two sides.
45.A ladder of length 13 m leans against a vertical wall. The foot of the ladder is 5 m from the base of the wall. Calculate how far up the wall the ladder reaches.
Medium
46.A right-angled triangle has legs of lengths 6 cm and 8 cm. What is the length of the hypotenuse?
Easy
- A11 cm
- B12 cm
- C9 cm
- D10 cm
47.A right-angled triangle has legs of lengths x cm and 24 cm, and hypotenuse 25 cm. What is x?
Medium
- A6 cm
- B7 cm
- C9 cm
- D8 cm
48.A right-angled triangle has hypotenuse 26 cm and one leg 10 cm. Find the length of the other leg.
Medium
- A22 cm
- B28 cm
- C26 cm
- D24 cm

Notes de révision

## Pythagoras Theorem - **Pythagoras** was a Greek mathematician (2500+ years ago). - **Pythagoras' theorem** links the three sides of a **right-angled triangle**. - The **hypotenuse** is the longest side, opposite the right angle. - Formula: **a² + b² = c²** where c is the hypotenuse, a and b are the shorter sides. ### Finding the Hypotenuse - Square the two shorter sides, add them, then take the positive square root. - c = √(a² + b²) ### Finding a Shorter Side - Square the hypotenuse and the other shorter side, subtract, then take the positive square root. - a = √(c² – b²) ### Tips - The hypotenuse must be the longest side. If it's not, check your working. - When finding a shorter side, subtract the smaller square from the larger to avoid a negative under the square root. - In multi-step problems, keep answers exact (e.g., √63) until the final step, then round. - Pythagoras can be used with any shape that can be split into right-angled triangles (e.g., rectangles, squares). ### Common Question Types - Find the hypotenuse given two shorter sides. - Find a shorter side given the hypotenuse and one side. - Find the diagonal of a rectangle or square. - Combine with other shapes (trapezium, multiple triangles). - Calculate perimeter or area after finding missing lengths. > **Example**: In a right triangle with legs 12 cm and 9 cm, hypotenuse = √(12²+9²) = √(144+81) = √225 = 15 cm.

Pythagoras' theorem: a² + b² = c², where c is the length of the hypotenuse and a, b are the lengths of the two shorter sides of a right-angled triangle.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Right-angled triangle: A triangle with one angle equal to 90°.

Cartes mémo

What is Pythagoras' theorem?

For a right-angled triangle, a² + b² = c², where c is the hypotenuse (longest side) and a, b are the two shorter sides.

How do you find the length of the hypotenuse using Pythagoras?

Square the two shorter sides, add them, then take the positive square root: c = √(a² + b²).

How do you find the length of a shorter side using Pythagoras?

Square the hypotenuse and the other shorter side, subtract the smaller square from the larger, then take the positive square root: a = √(c² - b²).

What is the hypotenuse?

The longest side of a right-angled triangle, opposite the right angle.

In a right triangle with legs 3 cm and 4 cm, find the hypotenuse.

c = √(3² + 4²) = √(9+16) = √25 = 5 cm.

In a right triangle with hypotenuse 13 cm and one leg 5 cm, find the other leg.

a = √(13² - 5²) = √(169-25) = √144 = 12 cm.

A flagpole is 25 m tall. A rope from its top is tied to the ground 8 m from its base. Find the rope length.

Rope length = √(25² + 8²) = √(625+64) = √689 ≈ 26.2 m.

A rectangle has sides 14 cm and w cm, diagonal 23 cm. Find w.

w = √(23² - 14²) = √(529-196) = √333 ≈ 18.2 cm.

Find the diagonal of a rectangle with sides 12 cm and 5 cm.

Diagonal = √(12² + 5²) = √(144+25) = √169 = 13 cm.

A rectangle has one side 12 cm and diagonal 13 cm. Find its area.

Other side = √(13² - 12²) = √(169-144) = √25 = 5 cm. Area = 12 × 5 = 60 cm².

A guy rope of length 6.9 m is attached to a flagpole top and fixed 3.2 m from its base. Find the flagpole height.

Height = √(6.9² - 3.2²) = √(47.61 - 10.24) = √37.37 ≈ 6.11 m.

In triangle ABC, AB = 12 cm, AD = 9 cm, AC = 22 cm, and BD is perpendicular to AC. Find BC (x).

BD = √(12² - 9²) = √63 ≈ 7.937 cm. DC = 22 - 9 = 13 cm. BC = √(BD² + DC²) = √(63+169) = √232 ≈ 15.2 cm.

A trapezium has AB=150 m, BC=90 m, CD=120 m, with right angles at B and C. Show AD ≈ 95 m.

Drop perpendicular from D to AB, forming a right triangle with legs (150-120)=30 m and 90 m. AD = √(30² + 90²) = √(900+8100) = √9000 ≈ 94.9 m, correct to nearest metre 95 m.

For the trapezium above, find the cost of a fence at $48 per 5-metre section.

Perimeter = AB+BC+CD+AD = 150+90+120+95 = 455 m. Number of sections = 455/5 = 91. Cost = 91 × $48 = $4368.

A right triangle has sides 5 cm, 12 cm, and 13 cm. Find its perimeter.

Perimeter = 5 + 12 + 13 = 30 cm.

A rectangular pond ABCD has AB = 12 m, diagonal AC = 15 m. Find BC.

BC = √(15² - 12²) = √(225-144) = √81 = 9 m.

For the pond above, find the length of fence around its edge.

Perimeter = 2(12+9) = 42 m.

What common mistake should you avoid when using Pythagoras?

Ensure the hypotenuse is the longest side; if it ends up shorter than another side, you made an error. When finding a shorter side, subtract the smaller square from the larger to avoid a negative under the square root.

How can Pythagoras be used with rectangles?

The diagonal splits the rectangle into two right-angled triangles, with the diagonal as the hypotenuse. So diagonal² = length² + width².

In a right triangle, if a = 6 and b = 8, find c.

c = √(6² + 8²) = √(36+64) = √100 = 10.

In a right triangle, if c = 17 and a = 15, find b.

b = √(17² - 15²) = √(289-225) = √64 = 8.

A 25 m flagpole has a rope from top to ground 8 m from base. Find rope length (exact form).

Rope = √(25² + 8²) = √(625+64) = √689 m.

A rectangle 14 cm by w cm has diagonal 23 cm. Find w (exact form).

w = √(23² - 14²) = √(529-196) = √333 cm.

A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.

c = √(9² + 12²) = √(81+144) = √225 = 15 cm.

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

Other leg = √(10² - 6²) = √(100-36) = √64 = 8 cm.

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