Question

Solve for x. Round your answer to the nearest tenth if necessary. MO is 13.7, OL is 10.3, NK is 12.9, and MN is x.

Answer 1

Interpreting M as the external point with two secants through the circle (one through O and L, the other through N and K), the secant-secant power theorem gives

MO · ML = MN · MK.

Compute ML = MO + OL = 13.7 + 10.3 = 24, and MK = MN + NK = x + 12.9. So

13.7(24) = x(x + 12.9)
328.8 = x^2 + 12.9x
x^2 + 12.9x - 328.8 = 0.

Solve the quadratic:
x = [−12.9 ± sqrt(12.9^2 + 4·328.8)]/2
= [−12.9 ± sqrt(1481.61)]/2
= [−12.9 ± 38.504]/2.

Positive root: x ≈ (25.604)/2 ≈ 12.802 → 12.8 (to the nearest tenth).

So x ≈ 12.8.