A tangent is drawn to a circle inscribed in triangle ABC. Crossing the sides of AC and BC at points P and Q

A tangent is drawn to a circle inscribed in triangle ABC. Crossing the sides of AC and BC at points P and Q, respectively. Find the perimeter of triangle CPQ if AB = 22 and the perimeter of triangle ABC is 78.

Segments AB and AC are tangent drawn from one point, then AD = AM.

BA and BC are tangents from point B, then by the property of tangents, BD = BH.

Then AM + BH = AB = 22 cm.

The perimeter of the triangle ABC = AB + BC + AC = AB + BH + CH + AM + CM = 78 cm.

Ravs = AB + (BH + CH) + (CM + CH) = 78.

(CM + CH) = 78 – 22 – 22 = 34 cm.

CM = CH as tangents drawn from one point, then CM = CH = 34/2 = 17 cm.

РМ = PК, QН = QК also by the property of tangents, then СН = СQ + QК, and СМ = СР + PК.

Then the perimeter of the triangle CQP will be equal to: P = CQ + QK + KP + CP = CH + CM = 17 + 17 = 34 cm.

Answer: The perimeter of the CPQ triangle is 34 cm.