From point A lying on the circle, two chords AB and AC are drawn. Points K and F are the midpoints of the chords

| February 11, 2021 | education

From point A lying on the circle, two chords AB and AC are drawn. Points K and F are the midpoints of the chords. Find the radius of the circle if AB = 9cm, AC = 17cm and KF = 5cm.

Let’s connect the edges of the chords AB and AC, points B and C. In the resulting triangle ABC, the segment KF is its midline, since points K and F are the midpoints of the chords AB and AC. Then the length of the BC side = 2 * KF = 2 * 5 = 10 cm.

Let’s define the semiperimeter of the triangle ABC. P = (AB + BC + AC) / 2 = (9 + 10 + 17) / 2 = 18 cm.

By Heron’s theorem, we determine the area of the triangle ABC.

S = √p * (p – AB) * (p – BC) * (p – AC) = √18 * 9 * 1 * 8 = √1296 = 36 cm2.

Determine the radius of the circumscribed circle.

R = AB * BC * AC / 4 * S = 9 * 10 * 17/4 * 36 = 1530/144 = 10.625 cm.

Answer: The radius of the circle is 10.625 cm.



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