A circle of radius r is inscribed in triangle ABC, touching the sides AB and BC at points K and M

| June 19, 2021 | education

A circle of radius r is inscribed in triangle ABC, touching the sides AB and BC at points K and M, respectively, and the sides of AC at point T, with AT = r. Find the area of the CME triangle if you know that r = 2 and TC = 10.

Let’s construct the radii of the circle to the points of tangency.

Since AT = r, by condition, then OK = ОT = r.

OT is perpendicular to AC, OK is perpendicular to AB, then ATOK is square, and triangle ABC is rectangular.

By the property of a tangent drawn from one point, CT = CM = 10 cm, AT = AK = 2 cm, BK = BM = X cm.

Then AC = 2 + 10 = 12 cm, AB = 2 + X cm.

Savs = AC * AB / 2 = 12 * (2 + X) / 2 = 12 + 6 * X.

Also, the area of ​​the triangle ABC is equal to:

Sас = r * p = 2 * (12 + 10 + X + 2 + X) / 2 = 24 + 2 * X.

12 + 6 * X = 24 + 2 * X.

4 * X = 12.

X = 3 cm.

Then BC = 10 + 3 = 13 cm.

SinB = AC / BC = 12/13.

Then Sqm = BK * BM * SinB / 2 = 3 * 3 * (12/13) / 2 = 54/13 = 4 (2/13) cm2.

Answer: The area of ​​the KBM triangle is 4 (2/13) cm2.



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