A rhombus is inscribed in an isosceles triangle so that one side of it lies on the base, and the other

| March 6, 2021 | education

A rhombus is inscribed in an isosceles triangle so that one side of it lies on the base, and the other on the side of the triangle. The side of the rhombus is 10 cm and the perimeter of the triangle is 75. Find the sides of the triangle.

Let the sides of an isosceles triangle be equal to X cm. AB = BC = X cm, the base is equal to Y cm, AC = Y cm.

Then the perimeter of the triangle is:

Ravs = 2 * X + Y = 75 cm.

Y = 75 – 2 * X. (1).

Triangles ABC and KВН are similar in two angles, since the angle B is common, KН is parallel to AM, and therefore AC, then the angle BAC = ВKН as the corresponding angles.

Then: Y / 10 = X / (X – 10).

Y = 10 * X / (X – 10). (2).

Let’s solve the system of two equations 1 and 2.

75 – 2 * X = 10 * X / (X – 10).

75 * X – 2 * X ^ 2 – 750 + 20 * X = 10 * X.

2 * X ^ 2 – 85 * X + 750 = 0.

Having solved the quadratic equation, we get: X1 = 12.5 cm, X2 = 30 cm.

X1 is not suitable, since in this case the base is equal to: 75 – 2 * 12.5 = 50 cm, a triangle with such dimensions cannot be built.

X2 = 30 cm, then Y = 75 – 2 * 30 = 15 cm.

Answer: The sides are 30 cm, the base is 15 cm.



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