A tangent and a secant are drawn from one point to the circle.

A tangent and a secant are drawn from one point to the circle. The tangent is longer than the inner and outer secant by 2 and 4 cm. Find the length of the secant.

Given: Circumference; ABC – secant; AD – tangent; AB = AD – 4; BC = AD – 2. Find AC-? Solution: Since AC = AB + BC we substitute into this equality AB = AD – 4 and BC = AD – 2, we get: AC = 2AD – 6. Knowing that if from one point the tangent (AD) and secant are drawn to the circle (AC), then the product of the entire secant by its outer part (AB) is equal to the square of the tangent, we get: AD ^ 2 = AC AB, AD ^ 2 = (2AD-6) (AD-4), AD ^ 2 -14 AD + 24 = 0; AD = 2 (not suitable, because AD-2 = BC, BC> 0), AD = 12 cm. Then AC = 2 · 12 -6 = 18 cm. Answer: 18 centimeters.