In a right-angled triangle, the point of tangency of the inscribed circle is divided by the hypotenuse

| August 16, 2021 | education

In a right-angled triangle, the point of tangency of the inscribed circle is divided by the hypotenuse into segments 5 cm and 12 cm long. Find the smaller leg of the triangle.

From the center O of the circle, draw segments to the points of tangency with the sides of the triangle.

By the property of a tangent drawn from one point, the segment CK = CM, BM = BH = 12 cm, AK = AH = 5 cm.

The length of the hypotenuse AB = AH + BH = 5 + 12 = 17 cm.

Let CK = CM = X cm

Then AC = 5 + X, BC = 12 + X.

By the Pythagorean theorem AB ^ 2 = AC ^ 2 + BC ^ 2.

172 = (5 + X) ^ 2 + (12 + X) ^ 2.

289 = 25 + 10 * X + X ^ 2 + 144 + 24 * X + X ^ 2.

2 * X ^ 2 + 34 * X – 120 = 0.

X ^ 2 + 17 * X – 60 = 0.

Let’s solve the quadratic equation.

X = CK = 3 cm.

Then AC = 5 + 3 = 8 cm.

Answer: The length of the smaller leg is 8 cm.



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