Multiple choice

Let ABC be a right-angled triangle with hypotenuse BC of length 20 cm. If AP is perpendicular to BC, then the maximum possible length of AP, in cm, is

  1. 8 2

  2. 6 2

  3. 5

  4. 10
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If the legs are a and b, then a^2 + b^2 = 20^2 and the altitude to the hypotenuse is ab/20. The product ab is maximized when a = b, giving ab = 200 and AP = 200/20 = 10 cm.

AI explanation

In a right-angled triangle, the altitude to the hypotenuse relates to the two segments it creates, so AP squared equals BP times PC. The sum of BP and PC equals the hypotenuse 20. To maximize AP, we maximize the product BP times PC, which occurs when BP and PC are equal, making each segment 10 cm. Therefore, the maximum altitude AP squared equals 10 times 10, so AP is 10 cm.

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