Triangle Rotating in Square

Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:

  1. 20/3
  2. 32/3
  3. 12
  4. 40/3
  5. 15

Show/Hide Solution

Our point P makes a lengthy journey, with many stops, before returning home. Some moves, which we call “a”, as from P to A, are 120° around a circle of radius 2. The length of an “a” move is (120/360) · (2 · 2) = 4/3.

Shorter moves, which we call “c”, as from B to C, are 30°; their length is (30/360) · (2 · 2) = /3.

When P lands in corners or on edges, it pauses briefly.

We have 8 “a” moves and 8 “c” moves. The total trip is 8(4/3) + 8(/3) = 40/3. The answer is (d).

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