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Trigonometry - Pythagoras' Theorem

Grade 10IGCSE

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

🔑Concepts

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Pythagoras' Theorem applies only 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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If the square of the longest side is equal to the sum of the squares of the other two sides, the triangle is right-angled (Converse of Pythagoras).

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

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In 3D shapes, Pythagoras' Theorem can be applied twice to find diagonals of cuboids or heights of pyramids.

📐Formulae

a2+b2=c2a^2 + b^2 = c^2 (where cc is the hypotenuse)

c=a2+b2c = \sqrt{a^2 + b^2}

a=c2−b2a = \sqrt{c^2 - b^2}

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} (Distance between two points)

💡Examples

Problem 1:

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

Solution:

c=52+122=25+144=169=13c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 cm

Explanation:

Since we are looking for the hypotenuse, we square both known sides, add them together, and take the square root of the result.

Problem 2:

A ladder of length 10 m leans against a vertical wall. The foot of the ladder is 6 m away from the base of the wall. How high up the wall does the ladder reach?

Solution:

h=102−62=100−36=64=8h = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 m

Explanation:

In this scenario, the ladder is the hypotenuse (c=10c=10) and the distance from the wall is one side (b=6b=6). We need to find the vertical height (aa), so we subtract the square of the known side from the square of the hypotenuse.

Problem 3:

Determine if a triangle with side lengths 7 cm, 24 cm, and 25 cm is right-angled.

Solution:

72+242=49+576=6257^2 + 24^2 = 49 + 576 = 625. Since 252=62525^2 = 625, then 72+242=2527^2 + 24^2 = 25^2.

Explanation:

By using the converse of Pythagoras' Theorem, we check if a2+b2=c2a^2 + b^2 = c^2. Because the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is right-angled.