切线与割线的点的幂

Problem

From external point $A$, a tangent $AL$ and a secant $AKE$ are drawn to circle $P$, with $AK = 12$ and $KE = 36$; find the tangent length $AL$.

Step 1: Find the whole secant $AE$

The secant length is the near part plus the outer part, so

$$AE = AK + KE = 12 + 36 = 48.$$

Step 2: Apply the tangent-secant formula

Using the power of a point relation,

$$AL^2 = AK \cdot AE = 12 \cdot 48 = 576.$$

Taking the square root gives

$$AL = \sqrt{576} = 24.$$

Answer

$$AL = 24$$