The center of the circumscribed circle divides the height of an isosceles triangle drawn to the base into segments

The center of the circumscribed circle divides the height of an isosceles triangle drawn to the base into segments, the smaller of which is 8 cm, the base of the triangle is 12 cm. Find the area of this triangle.

Let’s draw a segment OA, which is equal to the radius of the circumscribed circle.

Consider a right-angled triangle AOH, whose angle H is straight, since BH is the height of the triangle, leg OH = 8 cm, and leg AH = AC / 2 = 12/2 = 6 cm, since in an isosceles triangle the height coincides with the median.

Then, by the Pythagorean theorem:

OA ^ 2 = OH ^ 2 + AH ^ 2 = 8 ^ 2 + 6 ^ 2 = 64 + 36 = 100.

ОА = 10 cm.

Segment ОВ = ОА = 10 cm, since both segments are circular radii.

Then the height BH = OB + OH = 10 + 8 = 18 cm.

Determine the area of the triangle ABC.

S = AC * BH / 2 = 12 * 18/2 = 108 cm2.

Answer: The area of the triangle is 108 cm2.

One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.