# An isosceles triangle with a height drawn to the base is inscribed in a circle

An isosceles triangle with a height drawn to the base is inscribed in a circle 10 cm in diameter. Find the base of the triangle if the height is 2 cm.

Let’s draw the radius of the circle OC which is equal to half the diameter OC = D / 2 = 10/2 = 5 cm.

The segment ОВ is equal to the radius of the circle, and the segment ВН, by condition, is equal to 2 cm, then the segment ОН = ОВ – ВН = 5 – 2 = 3 cm.

Consider a right-angled triangle COH, in which the hypotenuse OC = R = 5 cm, and the leg OH = 3 cm, then by the Pythagorean theorem CH ^ 2 = OC ^ 2 – OH ^ 2 = 5 ^ 2 – 3 ^ 2 = 25 – 9 = sixteen.

CH = √16 = 4 cm.

Since triangle ABC is isosceles, the height of BH is also the median of the triangle, which means CH = AH = 4 cm.

Then the length of the base of the triangle is: AC = CH * 2 = 4 * 2 = 8 cm.

Answer: The base of the triangle is 8 cm.

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