Pythagoras' Theorem

Spec reference: Geometry and measures - Pythagoras' theorem and trigonometry
Key idea: Use Pythagoras' theorem to find missing sides in right-angled triangles.

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▶ Pythagoras' Theorem

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The theorem

In a right-angled triangle:

$$a^2+b^2=c^2$$

where $c$ is the hypotenuse (the longest side, opposite the right angle).

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Finding the hypotenuse

Example

Find the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm.

$$c^2=6^2+8^2=36+64=100$$$$c=\sqrt{100}=\boxed{{\displaystyle 10\text{ cm}}}$$

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Finding a shorter side

Example

The hypotenuse is 13 cm and one leg is 5 cm. Find the other leg.

$$a^2={13}^2-5^2=169-25=144$$$$a=\sqrt{144}=\boxed{{\displaystyle 12\text{ cm}}}$$

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Pythagoras in 3D

Apply Pythagoras twice to find diagonals in 3D shapes.

Example

Find the length of the space diagonal of a cuboid $3\times 4\times 12$ cm.

Step 1 - base diagonal: $d=\sqrt{3^2+4^2}=\sqrt{25}=5$ cm

Step 2 - space diagonal: $D=\sqrt{5^2+{12}^2}=\sqrt{169}=\boxed{{\displaystyle 13\text{ cm}}}$

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Pythagorean triples

Common sets of integers that satisfy $a^2+b^2=c^2$:

$3,4,5$ and $5,12,13$ and $8,15,17$

(and any multiples: $6,8,10$ or $9,12,15$ etc.)

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Exam tips

Watch out for

• Always identify the hypotenuse first - it is opposite the right angle
• To find a shorter side, subtract: $a^2=c^2-b^2$
• Leave answers as surds or to 1 d.p. unless told otherwise

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Test Yourself

Question 1 of 5

Legs 5 and 12. Hypotenuse?