Pythagoras theorem

Q: What is the Pythagoras theorem?

In a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse.

In a right-angled triangle, the longest side is opposite the right angle and is called the hypotenuse. Pythagoras' theorem says the square on the hypotenuse equals the sum of the squares on the other two sides.

Formulas

Pythagoras theorem (c is the hypotenuse)

a^{2} + b^{2} = c^{2}

Finding the hypotenuse

c = a^{2} + b^{2}

Finding a shorter side

a = c^{2} - b^{2}

Worked examples

Example 1. The two shorter sides of a right-angled triangle are 6 cm and 8 cm. Find the hypotenuse.

$c^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$ .
$c = 100 = 10$ .

Answer: 10 cm

Example 2. The hypotenuse of a right-angled triangle is 13 cm and one side is 5 cm. Find the other side.

$b^{2} = 1 3^{2} - 5^{2} = 169 - 25 = 144$ .
$b = 144 = 12$ .

Answer: 12 cm

Example 3. Is a triangle with sides 7 cm, 24 cm and 25 cm right-angled?

The longest side is 25: $2 5^{2} = 625$ .
$7^{2} + 2 4^{2} = 49 + 576 = 625$ .
The two are equal, so it is right-angled.

Answer: Yes

Example 4. A 10 m ladder reaches 8 m up a wall. How far is its foot from the wall?

$d^{2} = 1 0^{2} - 8^{2} = 100 - 64 = 36$ .
$d = 6$ .

Answer: 6 m

Common mistakes

Using the wrong side as the hypotenuse. It is always the longest side, opposite the right angle.
Forgetting the final square root and stopping at $c^{2}$ .
Adding the squares when you need a shorter side. For a shorter side, subtract.

Practice questions with answers

1. In a right-angled triangle, the side opposite the right angle is called the:

AHypotenuse
BBase
CAltitude
DMedian

Show answer

Answer: A) Hypotenuse

The side opposite the right angle is the longest side and is called the hypotenuse.

2. In a right-angled triangle, what is the side opposite the right angle called?

AThe median
BThe base
CThe hypotenuse
DThe height

Show answer

Answer: C) The hypotenuse

The side opposite the right angle is the hypotenuse. It is always the longest side.

3. Which of the following is a Pythagorean triple?

A(5, 12, 13)
B(2, 3, 4)
C(4, 5, 6)
D(6, 7, 8)

Show answer

Answer: A) (5, 12, 13)

5^{2} + 1 2^{2} = 25 + 144 = 169 = 1 3^{2}

, so (5, 12, 13) is a Pythagorean triple.

4. A square sheet has a diagonal of 14 cm folded across it.
What is the area of the square sheet?

A98
B88
C108
D196

Show answer

Answer: A) 98

If side =

s

, diagonal

d

satisfies

d^{2} = s^{2} + s^{2} = 2 s^{2}

, so

s^{2} = \frac{d ^{2}}{2} = \frac{196}{2} = 98

. Area =

s^{2} = 98

sq cm.

5. The sides of a right triangle are

x, x + 1

and 5, where 5 is the hypotenuse. What is

x

Show answer

Answer: B) 3

Try values:

3^{2} + 4^{2} = 25 = 5^{2}

. So

x = 3. (x = 4

gives

16 + 25 = 41.

)

6. The perimeter of a right triangle is 24 cm and its hypotenuse is 10 cm.
Find the area of the triangle, in square centimetres.

Show answer

Answer: A) 24

Legs

a + b = 24 - 10 = 14

;

a^{2} + b^{2} = 100. (a + b)^{2} = a^{2} + 2 ab + b^{2} - > 196 = 100 + 2 ab - > ab = 48

. Area =

\frac{1}{2}

ab = 24

sq cm.

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Frequently asked questions

What is the Pythagoras theorem?

In a right-angled triangle, $a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse.

What are Pythagorean triples?

Sets of whole numbers that satisfy the theorem, such as 3, 4, 5 and 5, 12, 13 and 8, 15, 17. Multiples of them work too (6, 8, 10).

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

Square the three sides. If the square of the longest equals the sum of the other two squares, the triangle has a right angle.

Does the theorem work for any triangle?

No, only for right-angled triangles.

Pythagoras theorem

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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