Mathematics • Year 8 • Unit 3 • Lesson 2

Missing Legs in the Real World

Use a = √(c² − b²) for ladders, tent poles, kite strings, conveyor belts and a sailing boat's mast. Then explain your thinking in your own words.

Apply · Real-World Maths

1. Word problems

Each problem gives you the hypotenuse and one leg — you have to subtract to find the missing leg. Draw, label, subtract, square-root, verify. Show all working.

1.1 — The ladder problem. A 10 m ladder leans against a wall. The base of the ladder is 6 m from the wall.

(a) Sketch the ladder, wall and ground as a right triangle.
(b) How high up the wall does the ladder reach?
(c) Identify the Pythagorean triple involved. 3 marks

Stuck? The ladder is the hypotenuse (10 m); the gap on the ground is one leg (6 m); the wall height is the missing leg. a² = 100 − 36.

1.2 — Kite string. Maria flies a kite on 25 m of string. The kite is directly above a point 7 m away on the ground.

(a) Sketch the situation as a right triangle.
(b) How high above the ground is the kite?
(c) Spot the Pythagorean triple. 3 marks

Stuck? String = 25 (hypotenuse), horizontal = 7 (leg), height = missing leg. a² = 625 − 49 = 576. Triple: 7-24-25.

1.3 — Sailing boat mast. A sailing boat's mast is supported by a wire that runs from the top of the mast to a point on the deck 5 m from the base of the mast. The wire is 13 m long.

(a) Draw the right triangle formed by the mast, deck and wire.
(b) How tall is the mast?
(c) Verify your answer by checking a² + b² = c². 3 marks

Stuck? Wire = 13 (hypotenuse), deck distance = 5 (leg), mast = missing leg. a² = 169 − 25 = 144. (5-12-13 triple.)

1.4 — Slide at the playground. A slide is 4.5 m long (this is the slant side). The top of the slide is 2.7 m above the ground.

(a) Sketch the slide, ground, and ladder-back as a right triangle.
(b) How far along the ground (horizontally) does the slide reach? 3 marks

Stuck? Slide = 4.5 (hypotenuse), height = 2.7 (one leg), ground reach = missing leg. a² = 20.25 − 7.29. (Hint: 2.7-3.6-4.5 = 3-4-5 × 0.9.)

1.5 — Conveyor belt. A factory conveyor belt is 17 m long and slopes upward from the floor to a loading dock 8 m high.

(a) How far horizontally does the belt cover along the ground?
(b) Identify the Pythagorean triple. 3 marks

Stuck? Belt = 17 (hypotenuse), height = 8 (leg), horizontal distance = missing leg. a² = 289 − 64 = 225. (8-15-17 triple.)

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate is asked to find a missing leg, given c = 10 and b = 6. They write "a = √(10² + 6²) = √136 ≈ 11.66 cm". In your own words, explain (i) what mistake they made, (ii) what the correct working and answer are, and (iii) one sanity check based on the triangle's geometry that should have warned them their answer was impossible. Use the phrase "the hypotenuse must be the longest side" somewhere in your answer.

Stuck? Revisit lesson § Card 4 — finding a leg uses subtraction, not addition. Their answer (11.66) is BIGGER than the hypotenuse 10 — impossible for a leg.

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What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Ladder problem

(a) Right triangle: hypotenuse = ladder 10 m, base leg = 6 m, vertical leg = wall height.
(b) a² = 10² − 6² = 100 − 36 = 64, so a = 8 m up the wall.
(c) 6-8-10, which is 3-4-5 × 2.

1.2 — Kite string

(a) Right triangle: string = hypotenuse (25 m), horizontal gap = 7 m, height = missing leg.
(b) a² = 25² − 7² = 625 − 49 = 576, so kite is 24 m high.
(c) 7-24-25 Pythagorean triple.

1.3 — Sailing boat mast

(a) Right triangle: wire = hypotenuse (13 m), deck = 5 m, mast = missing leg.
(b) a² = 13² − 5² = 169 − 25 = 144, so mast = 12 m tall.
(c) Verify: 12² + 5² = 144 + 25 = 169 = 13² ✓ (5-12-13 core triple.)

1.4 — Slide

(a) Right triangle: slide = hypotenuse (4.5 m), vertical height = 2.7 m, ground reach = missing leg.
(b) a² = 4.5² − 2.7² = 20.25 − 7.29 = 12.96, so ground reach = √12.96 = 3.6 m. (2.7-3.6-4.5 = 3-4-5 × 0.9.)

1.5 — Conveyor belt

(a) a² = 17² − 8² = 289 − 64 = 225, so horizontal distance = 15 m.
(b) 8-15-17 Pythagorean triple.

2.1 — Explain your thinking (sample response)

The classmate ADDED instead of subtracting. When you know the hypotenuse and one leg, you must rearrange c² = a² + b² to a² = c² − b², so the correct working is a² = 10² − 6² = 100 − 36 = 64, giving a = √64 = 8 cm. A quick sanity check: the hypotenuse must be the longest side of a right triangle, so any leg must be SHORTER than 10 — but their answer (11.66 cm) is longer than the hypotenuse, which is geometrically impossible. That alone tells you they made an error and need to swap their + for a −.

Marking: 1 mark for identifying "added instead of subtracted"; 1 mark for the correct working and a = 8; 1 mark for the "leg cannot be longer than hypotenuse" sanity check; 1 mark for clear sentences using "the hypotenuse must be the longest side".