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Trigonometry - Pythagoras’ Theorem

Grade 12IGCSE

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

🔑Concepts

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Pythagoras’ Theorem applies exclusively 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 Converse of Pythagoras: If the sum of the squares of the two shorter sides equals the square of the longest side, the triangle must be right-angled.

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3D Pythagoras: The theorem can be extended to find the distance between two points in 3D space or the diagonal of a cuboid.

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Relationship to Trigonometric Identities: Pythagoras’ Theorem is the foundation for the identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.

📐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}

d2=x2+y2+z2d^2 = x^2 + y^2 + z^2 (Diagonal of a cuboid with dimensions x,y,zx, y, z)

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

💡Examples

Problem 1:

A ladder of length 13m leans against a vertical wall. The foot of the ladder is 5m away from the base of the wall. How high up the wall does the ladder reach?

Solution:

12m

Explanation:

Identify the hypotenuse (c=13c = 13) and one side (b=5b = 5). Using a2=c2−b2a^2 = c^2 - b^2, we get a2=132−52=169−25=144a^2 = 13^2 - 5^2 = 169 - 25 = 144. Taking the square root, a=144=12a = \sqrt{144} = 12.

Problem 2:

Calculate the length of the internal diagonal of a cuboid with dimensions 3cm, 4cm, and 12cm.

Solution:

13cm

Explanation:

Using the 3D Pythagoras formula d2=l2+w2+h2d^2 = l^2 + w^2 + h^2. Here, d2=32+42+122=9+16+144=169d^2 = 3^2 + 4^2 + 12^2 = 9 + 16 + 144 = 169. Thus, d=169=13d = \sqrt{169} = 13.

Problem 3:

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

Solution:

Yes, it is right-angled.

Explanation:

Check if a2+b2=c2a^2 + b^2 = c^2. Calculate 72+242=49+576=6257^2 + 24^2 = 49 + 576 = 625. Calculate 252=62525^2 = 625. Since 625=625625 = 625, the converse of Pythagoras' theorem confirms it is a right-angled triangle.