Let $$A0$$,$$1$$,$$B1$$,$$1$$, $$C1$$,−$$1$$, D−$$1$$,$$0$$ be four points. If P is any other point, then $$PA+PB+PC+PD$$≥d, when d is $$___$$, where [.] represents greatest integer.

Cartesian plane

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Question

Let $A (0, 1), B (1, 1), C (1, - 1), D (- 1, 0)$ be four points. If P is any other point, then $PA + PB + PC + PD \geq d,$ when $[d]$ is ___, where [.] represents greatest integer.

Moderate

Solution

$(AP + PC)$ is minimized when P lies on AC and BP+PD is minimized when P lies on BD $AP + PB + PC + PD + = AC + BD = 2 \sqrt{5}$

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