Mathematics
Pink Kangaroo
2023
11

11

The Pentagon PQRST is divided into four triangles with equal perimeters. The triangle PQR is equilateral. PTU, SUT and RSU are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon PQRST to the perimeter of the triangle PQR?

A. 2 : 1 B. 3 : 2 C. 4 : 3 D. 5 : 3 E. 5 : 2

Answer

D.

SOLUTIOn

Let 2a be the side of equilateral triangle PQR. Then PQR has a perimeter of 6a and PU has a length a. Therefore, in order for the perimeter of triangle PTU to be 6a, PT must be 5 2 a. Since the isosceles triangles are congruent ST = PU and RS = PT. So the perimeter of the pentagon PQRST is 2a + 2a + 5/2 a + a + 5/2 a = 10a. Therefore the ratio wanted is 10a : 6a, that is 5 : 3.

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