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M is the midpoint of side AB of an equilateral triangle ABC, where AB = 20. P is a point on side BC such that the total distance AP + PM is minimised. Which of the following is AP + PM greater than?

1. 10*sqrt(7)
2. 10*sqrt(3)
3. 10*sqrt(5)
4. 10

Correct answer: 10*sqrt(7)

Solution

By reflecting M across BC (or using coordinate geometry with AB=20), the minimum value of AP+PM equals sqrt(700) = 10*sqrt(7), so AP+PM is greater than 10*sqrt(7) is the tightest bound among the options.

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