Chords AC and diameter AB are drawn through point A of the circle. from the vertex С of the triangle ABC

Chords AC and diameter AB are drawn through point A of the circle. from the vertex С of the triangle ABC the height СD is drawn. Find the diameter of the circle if AD = 27 cm and the chord is 45 cm.

The ACD triangle is rectangular, since the CD is the height drawn to the diameter of the circle.

By the Pythagorean theorem, we find the length of the leg CD.

CD ^ 2 = AC ^ 2 – AD ^ 2 = 45 ^ 2 – 27 ^ 2 = 2025 – 729 = 1296.

СD = 36 cm.

Since the inscribed angle ACB is based on the diameter of the circle, this angle is 90, and therefore the triangle ABC is rectangular.

In a right-angled triangle, the square of the height drawn to the hypotenuse is equal to the product of the lengths of the segments by which the height divides the hypotenuse.

CD ^ 2 = AD * DB.

36 ^ 2 = 27 * DB.

DB = 1296/27 = 48 cm.

Then the diameter of the circle will be equal to: AB = AD + DB = 27 + 48 = 75 cm.

Answer: The diameter of the circle is 75 cm.