The side of an isosceles triangle is divided by the point of tangency of the inscribed circle in a ratio of 3: 8

| February 5, 2021 | education

The side of an isosceles triangle is divided by the point of tangency of the inscribed circle in a ratio of 3: 8, counting from the apex of the angle at the base of the triangle. Find the base of the triangle if its perimeter is 56 cm.

Let us designate the triangle given by the condition ABC, AB = BC. Points H, K, M – points of tangency of a circle with sides AB, BC, AC.
In solving the problem, we use the property of the lateral sides of an isosceles triangle and the theorem on tangents drawn from one point.
We enter the proportionality coefficient x and write down the segments:
AH = 3x, BH = 8x.
СK = 3x, ВK = 8x.
AM = AH = 3x.
CM = СK = 3x.
The perimeter of the triangle is known by the condition, we make the equation:
3x + 8x + 3x + 8x + 3x + 3x = 56
28x = 56
x = 2
AC = AM + CM = 3x + 3x = 6x = 12 (cm).
Answer: the base of the triangle is 12 cm.



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