The two sides of an acute-angled triangle are 13 cm and 15 cm. The height drawn to the third side is 12 cm

| September 11, 2021 | education

The two sides of an acute-angled triangle are 13 cm and 15 cm. The height drawn to the third side is 12 cm. Find the radii of the inscribed and circumscribed circles of the triangle.

Let’s designate our triangle ABC, by condition we know: AB = 13 cm, BC = 15 cm, height BH = 12 cm.
To find the radii of the inscribed and circumscribed circles, you need the third side of the AC triangle.
From the right-angled triangle АНB we find АН:
AH = √ (AB² – BH²) = √ (169 – 144) = √25 = 5 (cm).
From the right-angled triangle СНB we find СН:
CH = √ (BC² – BH²) = √ (225 – 144) = √81 = 9 (cm).
AC = AH + CH = 5 + 9 = 14 (cm).
Let’s find the area of ​​the triangle ABC:
S = 1/2 * AC * BH = 1/2 * 14 * 12 = 84 (cm²).
The radius of the circumscribed circle: R = a * b * c / 4S = 13 * 15 * 14/4 * 84 = 2730/336 = 8.125 cm.
Inscribed circle radius: r = 2S / (a ​​+ b + c) = 2 * 84 / (13 + 15 + 14) = 168/42 = 4 cm.
Answer: R = 8.125 cm, r = 4 cm.



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