### Question 13

SSC-CGL 2018 June 4 Shift 1

A circle is drawn inside a triangle ABC. The circle touches the sides AB, BC and AC at the points R, P and Q respectively. If AQ = 4.5 cm, PC = 5.5 cm and BR = 6 cm, then the perimeter of triangle ABC is:

- 32 cm
- 28 cm
- 30.5 cm
- 26.5 cm

### Solution 1

Use the theorem that tangents to a circle from an external point are equal in length.

Observe that AR and AQ will be equal tangents measuring 4.5 cm each. Similarly, CQ and CP each = 5.5 cm. And BR and BP each equal to 6 cm.

Adding all the above lengths, perimeter is 4.5x2 + 5.5x2 + 6x2 = 32 cm Ans!

Any other method? Let me know in the comments.

EXPLANATION: triangle ABC contains a circle completely inscribed in it. The circle touches its sides at P, Q and R. Observe that we can view the whole figure from each of the vertices A, B and C. From the vertex A there are two tangents to the circle AR and AQ, both of which are equal because two tangents from an external point are equal. The lengths AQ, PC and BR have been given as 4.5, 5.5 and 6 centimeters. From these lengths we can find the pairing tangents, and hence the perimeter.

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