From a point outside the circle, a secant is drawn, intersecting the circle at points 12 cm and 20 cm

| February 28, 2021 | education

From a point outside the circle, a secant is drawn, intersecting the circle at points 12 cm and 20 cm away from this one. The distance from this point to the center of the circle is 17 cm. Find the radius of the circle.

In triangles ACO and ABO, according to the cosine theorem, we express the values of the radii OB and OC, which are the radii of the circle.

R ^ 2 = AC ^ 2 + AO ^ 2 – 2 * AC * AO * CosA = 400 + 289 – 2 * 20 * 17 * СosA = 689 – 680 * CosA. (one).

R ^ 2 = AB ^ 2 + AO ^ 2 – 2 * AB * AO * CosA = 144 + 289 – 2 * 12 * 17 * CosA = 433 – 408 * CosA.

Let us equate the right-hand sides of the equalities.

689 – 680 * CosA = 433 – 408 * CosA

272 * CosA = 256.

CosA = 256/272 = 16/17.

Substitute the CosA value into Equation 1.

R ^ 2 = 689 – 680 * 16/17 = 689 – 640 = 49

R = 7 cm.

Answer: The radius of the circle is 7 cm.



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