Geometry - Pythagoras' Theorem

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Pythagoras' Theorem only applies to right-angled triangles.

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The hypotenuse is the longest side of a right-angled triangle and is always opposite the 90-degree angle.

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The square of the hypotenuse is equal to the sum of the squares of the other two sides.

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a^2 + b^2 = c^2

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Pythagorean Triples are sets of three integers that satisfy the theorem (e.g., 3, 4, 5 or 5, 12, 13).

📐Formulae

a^2 + b^2 = c^2

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

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

💡Examples

Problem 1:

A right-angled triangle has two shorter sides of length 5 cm and 12 cm. Calculate the length of the hypotenuse.

Solution:

c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

Explanation:

To find the hypotenuse, square both shorter sides, add them together, and then take the square root of the result.

Problem 2:

In a right-angled triangle, the hypotenuse is 10 cm and one side is 6 cm. Find the length of the third side.

Solution:

a = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8

Explanation:

When the hypotenuse and one side are known, subtract the square of the known side from the square of the hypotenuse, then take the square root.

Problem 3:

A ladder of length 5 m leans against a vertical wall. The foot of the ladder is 2 m away from the wall. Calculate how far up the wall the ladder reaches, giving your answer to 2 decimal places.

Solution:

h = \sqrt{5^2 - 2^2} = \sqrt{25 - 4} = \sqrt{21} \approx 4.58

Explanation:

The ladder acts as the hypotenuse (5m) and the distance from the wall is one side (2m). We solve for the height (the other side) and round to the required precision as per IGCSE standards.