In an isosceles triangle, the base is 10, and the side is 13 cm. Find the r inscribed in it and R

In an isosceles triangle, the base is 10, and the side is 13 cm. Find the r inscribed in it and R in the circumscribed circle around it.

We find the semiperimeter of the given triangle by the condition:
p = 1/2 * (a + b + c) = 1/2 * (10 + 13 + 13) = 18 (cm).
We find the radius of the inscribed circle by the formula:
r = √ (((p – a) * (p – b) * (p – c)) / p) = √ (((18 – 10) * (18 – 13) * (18 – 13)) / 18 ) = √ ((8 * 5 * 5) / 18) = √ (200/18) = 10√2 / 3√2 = 10/3 = 3.333333 ≈ 3.33 (cm).
The radius of the circumscribed circle is found by the formula:
R = a * b * c / 4 * √ (p * (p – a) * (p – b) * (p – c)) = 1690/4 * √ (18 * 8 * 5 * 5)) = 1690 / 240 = 7.0416666 ≈ 7.04 (cm).
Answer: r = 3.33 cm, R = 7.04 cm.