The length of a circle circumscribed around an isosceles triangle is 50 cm.

The length of a circle circumscribed around an isosceles triangle is 50 cm. Find the perimeter of the triangle if the height to the base is 32 cm.

Find the radius of the circle along the known length:

50P = 2Pr;

r = 25.

The center of the circle is at the intersection of the mid-perpendiculars. This perpendicular will be the height BE dropped to the base of the speaker. The center of the circle will be at point D at a distance equal to the radius r from the vertex B:

BD = r = 25;

DE = BE – BD = 32 – 25 = 7.

Find the side AE of right triangle ADE:

AE = √ (r ^ 2 – DE ^ 2) = √ (25 ^ 2 – 7 ^ 2) = 24;

AC = 2 * AE = 2 * 24 = 48.

Side AB of right triangle ABE:

AB = √ (AE ^ 2 + BE ^ 2) = √ (24 ^ 2 + 32 ^ 2) = 40;

BC = AB = 40;

Perimeter p:

p = AB + BC + AC = 40 + 40 + 48 = 128.

Answer: 128.