# 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.

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