The height drawn to the base of the isosceles triangle is 9 cm, and the base itself is 24 cm. Find the radii of the circles inscribed in the triangle and circumscribed about the triangle.
Since the triangle is isosceles, the height BH divides the base of the AC into equal segments. AH = CH = AC / 2 = 24/2 = 12 cm.
From the right-angled triangle ABH, by the Pythagorean theorem, we define the hypotenuse AB.
AB ^ 2 = AH ^ 2 + BH ^ 2 = 12 ^ 2 + 9 ^ 2 = 144 + 81 = 225.
AB = BC = 15 cm.
Let us determine the area and semiperimeter of the triangle.
Sас = АС * BН / 2 = 24 * 9/2 = 108 cm2.
p = (AB + BC + AC) / 2 = (15 + 15 + 24) / 2 = 27 cm.
Then the radius of the inscribed circle is: r = S / p = 108/27 = 4 cm.
Determine the radius of the circumscribed circle.
R = (AB * BC * AC) / (4 * S) = 15 * 15 * 24/4 * 108 = 75/8 = 12.5 cm.
Answer: The radius of the inscribed circle is 4 cm, the radius of the inscribed circle is 12.5 cm.
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