The lateral side of an isosceles triangle, in which the circle is inscribed, is 50 cm. The height of the same triangle

| February 6, 2021 | education

The lateral side of an isosceles triangle, in which the circle is inscribed, is 50 cm. The height of the same triangle, drawn to the base, is 40cm. Find the distance between the points of tangency of the circle with the sides of the triangle.

Consider a right-angled triangle ABН, in which the angle AHB is straight, AB = 50 cm, BH = 40 cm.

Then, by the Pythagorean theorem, the leg AН will be equal to:

AH ^ 2 = AB ^ 2 – BH ^ 2 = 2500 – 1600 = 900. AH = 30 cm.Then AC = 2 * AH = 60 cm.

By the property of tangents, the segments of tangents drawn from one point are equal, which means AK = AH = 30 cm, then the segment KB = AB – AK = 50 – 30 = 20 cm.

Consider triangles ABC and ВKM.

Let us prove that they are similar.

So ВK = ВM, by the property of tangents, and AB = BC, since ABC is isosceles, then the segments AB, BC are similar to the segments ВK and ВM. Angle B is common for both triangles, which means ABC is similar to КВМ in the second characteristic.

Then AB / KВ = AC / KM.

50/20 = 60 / KM.

KM = 20 * 60/50 = 24 cm.

Answer: KM = 24 cm.



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