A straight ladder is leaning against a vertical wall, with its foot resting on horizontal ground. The ladder has length 4.5 m and the top of the ladder touches the wall at a point 4.1 m vertically above the ground. (a) Calculate the distance from the foot of the ladder to the ba…

(a) 1.85 m (b) AB = √(74) ≈ 8.6 km

Pythagoras' Theorem: O-Level / IGCSE Maths (0580)

Syllabus C6.1, E6.1 · Strand 6 Trigonometry

Questions: 3
Total marks: 14
Tier mix: 2 Core · 1 Extended

Pythagoras’ theorem describes the relationship between the three sides of any right-angled triangle. If the two shorter sides (the legs) have lengths $a$ and $b$ , and the longest side opposite the right angle (the hypotenuse) has length $c$ , then $a^2 + b^2 = c^2$ . This topic focuses on using that relationship to calculate a missing side length when the other two sides are known, a skill that underpins much of the trigonometry strand and recurs throughout later geometry work.

The key method is to first identify which side is the hypotenuse, since it is always opposite the right angle and is the longest side. To find the hypotenuse, square both legs, add them, and take the positive square root: $c = \sqrt{a^2 + b^2}$ . To find one of the shorter sides instead, rearrange to subtract before taking the root: $a = \sqrt{c^2 - b^2}$ . Setting out each line clearly and keeping an unrounded value until the final step helps avoid accuracy errors, and answers should carry appropriate units and sensible rounding. Exam-style questions often embed the triangle in a 2D context, such as a ladder against a wall, a diagonal across a rectangle, or the distance between two points, so part of the work is recognising where the right angle lies and sketching or labelling the triangle before substituting.

The worked examples below are original, written to match the syllabus objective of applying Pythagoras’ theorem to find a missing side of a right-angled triangle, including in 2D problem contexts. Each one shows a full worked solution so you can follow the reasoning step by step.

Question 1

Structured Core 6 marks

A straight ladder leans against a vertical wall, with its foot resting on horizontal ground. The wall meets the ground at a right angle.

The foot of the ladder is 1.4 m from the base of the wall, and the top of the ladder reaches 4.8 m up the wall.

(a) Calculate the length of the ladder. [3]

(b) A guy rope is used to support a vertical flagpole standing on horizontal ground. One end of the rope is fixed to the top of the pole, 4 m above the ground, and the other end is fixed to a peg in the ground 2.5 m from the foot of the pole. The pole is at right angles to the ground.

Calculate the length of the guy rope. Give your answer correct to 2 decimal places. [3]

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Question 2

Structured Extended 6 marks

A straight ladder is leaning against a vertical wall, with its foot resting on horizontal ground.

The ladder has length $4.5\text{ m}$ and the top of the ladder touches the wall at a point $4.1\text{ m}$ vertically above the ground.

(a) Calculate the distance from the foot of the ladder to the base of the wall, measured along the ground. Give your answer correct to 2 decimal places. [3]

A surveyor records two markers on a coordinate grid, where each unit represents $1\text{ km}$ . Marker $A$ is at the point $(-3,\ 2)$ and marker $B$ is at the point $(4,\ -3)$ .

(b) Calculate the straight-line distance $AB$ . Give your answer in kilometres, correct to 1 decimal place. [3]

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Question 3

Multiple choice Core 2 marks

A straight support cable runs from the top of a vertical flagpole to a fixing point on level ground. The cable is $17\text{ m}$ long, and the fixing point on the ground is $8\text{ m}$ from the base of the pole. The pole meets the ground at a right angle.

What is the height of the flagpole?

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