In an isosceles triangle, the bisector of the angle at the base divides the lateral side into two segments

In an isosceles triangle, the bisector of the angle at the base divides the lateral side into two segments of 30 cm and 25 cm, starting from the apex at the base. Find the radius.

Let us introduce the designation of the sides of this triangle, x is the lateral side, y is the base of the isosceles triangle.
Find the side:
x = 30 + 25 = 55 (cm).
Based on the property of the bisector of a triangle, we write the ratio:
25/30 = x / y;
25/30 = 55 / y.
Let us express and find y – base:
y = 30 * 55/25 = 1650/25 = 66 (cm).
We find the radius of the circumscribed circle by the formula:
R = x² / √ ((2x) ² – y²) = 55² / √ (110² – 66²) = 3025/88 = 34.375 (cm).
Answer: the radius of the circumscribed circle is 34.375 cm.