# In a triangle ABC AB = 10 AC = 12. Perimeter of triangle ABC = 32. 1) Find the radius of a circle circumscribed about

**In a triangle ABC AB = 10 AC = 12. Perimeter of triangle ABC = 32. 1) Find the radius of a circle circumscribed about a triangle. 2) determine the type of triangle by the lengths of its sides. 3) find the area of the triangle. 4) find the height dropped from vertex B.**

Let us determine the length of the BC side.

BC = Ravs – AB – AC = 32 – 10 – 12 = 10 cm.

Since AB = BC, the triangle ABC is isosceles with the base AC.

Let us define the area of the triangle ABC by Heron’s theorem. The semi-perimeter of the triangle is: p = P / 2 = 32/2 = 16 cm.

Then S = √16 * (16 – 10) * (16 – 10) – (16 – 12) = √16 * 6 * 6 * 4 = √2304 = 48 cm2.

Determine the radius of the circumscribed circle.

R = AB * BC * AC / 4 * S = 10 * 10 * 12/4 * 48 = 6.25 cm

The area of the triangle ABC is also equal to: Sавс = АС * ВН / 2.

VN = 2 * Savs / AC = 2 * 48/12 = 8 cm.

Answer: The radius is 6.25 cm, triangle ABC is isosceles, the area is 48 cm2, the height is 8 cm.