2023 · Q59
The hypotenuse AC of a right-angled triangle ABC is 3√5 cm. If AB is doubled and BC is tripled such that ABC remains a right-angled triangle, the hypotenuse becomes 15 cm. What is AB + BC equal to?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A10 cm
- B9 cm
- C2√5 cm
- D8 cm
Explanation
Let the right angle be at B, so AB² + BC² = AC² = (3√5)² = 45. After doubling AB and tripling BC the triangle is still right-angled with hypotenuse 15, so (2AB)² + (3BC)² = 15² = 225, i.e. 4AB² + 9BC² = 225. Writing u = AB² and v = BC² gives u + v = 45 and 4u + 9v = 225; substituting u = 45 − v yields 180 + 5v = 225, so v = 9 and u = 36. Thus AB = 6 and BC = 3, and AB + BC = 9 cm. Hence option (b). The distractor 2√5 cm is just half the original hypotenuse, and 8 or 10 cm fail the scaled Pythagoras check.