# Find the radius of a circle inscribed in an acute-angled triangle ABC if the height of the triangle

**Find the radius of a circle inscribed in an acute-angled triangle ABC if the height of the triangle is BH = 12 and it is known that the sine A = 12/13, the sine C = 4/5.**

In a right-angled triangle ABN, AB = ВН / SinA = 12 / (12/13) = 13 cm.

In a right-angled triangle ВСН, ВС = ВН / SinC = 12 / (4/5) = 15 cm.

By the Pythagorean theorem, AH ^ 2 = AB ^ 2 – BH ^ 2 = 169 – 144 = 25.

AH = 5 cm.

By the Pythagorean theorem, CH ^ 2 = BC ^ 2 – BH ^ 2 = 225 – 144 = 91.

CH = 9 cm.

Then AC = AH + CH = 5 + 9 = 14 cm.

Determine the area of the triangle ABC. Savs = AC * ВН / 2 = 14 * 12/2 = 84 cm2.

Let us define the semi-perimeter of the triangle: p = (13 + 15+ 14) / 2 = 21 cm.

Then the radius of the inscribed circle is: r = S / p = 84/21 = 4 cm.

Answer: The radius of the circle is 4 cm.