A triangle ABC has been divided into four smaller triangles P, Q, R, S whose perimeters are 16 cm, 12 cm, 4 cm and 12 cm respectively. P, R and S contain the vertices A, B and C respectively. What is the perimeter of the triangle ABC?

Explanation

The triangle ABC is divided by the central triangle Q. Let the vertices of Q be D, E, F on sides AB, BC, CA respectively. Thus, P is \triangle ADF, R is \triangle BDE, and S is \triangle CEF. The sum of the perimeters of P, R, and S is (AD+DF+FA) + (BD+DE+EB) + (CE+EF+FC) = (AD+BD) + (BE+EC) + (CF+FA) + (DF+DE+EF) = \text{Perimeter of ABC} + \text{Perimeter of Q}. Given P=16, R=4, S=12 and Q=12. So, 16+4+12 = \text{Perimeter of ABC} + 12 \implies \text{Perimeter of ABC} = 32 - 12 = 20 cm.

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