In the triangle ABC, AB = 10 cm, BC = 12 cm, CA = 5 cm, the inscribed circle of which

In the triangle ABC, AB = 10 cm, BC = 12 cm, CA = 5 cm, the inscribed circle of which touches the staron AB, BC, CA at points PQR. Find AR, PB, BQ. QC. CR. RA

By the property of tangents, the lengths of the segments of tangents to the circle drawn from one point are equal.

Then, AP = AR, CR = CQ, BQ = BP.

Let the length of the segment AR = AP = X cm, then:

PB = AB – X = 10 – X, and CR = AC – X = 5 – X.

Segment CB = CQ + DQ, since CQ = CR, and BQ = BP, then CB = CR + PB = (5 – X) + (10 – X) = 12.

15 – 2 * X = 12.

2 * X = 15 – 12.

X = 3/2 = 1.5 cm.

AR = AP = 1.5 cm, then:

PB = BQ = 10 – 1.5 = 8.5 cm,

CR = CQ = 5 – 1.5 = 3.5 cm.

Answer: AR = AP = 1.5 cm, PB = BQ = 8.5 cm, CR = CQ = 3.5 cm.