Questions: If AC=12, BC=3, find CE

Transcript text: If $\overline{\mathrm{AC}}=12, \overline{\mathrm{BC}}=3$, find $\overline{\mathrm{CE}}$

Solution

Solution Steps

Step 1: Power of a Point Theorem

The power of a point theorem states that for a point C outside a circle and two secant lines CA and CE intersecting the circle at points A, B, and D, E, respectively, the product of the lengths of the segments from C to the two intersection points of each line is the same: CA * CB = CD * CE.

Step 2: Substituting given values

We are given AC = 12 and BC = 3. Also, note that CD = CB + BD and CE = BC + BE, but the lengths of BD and BE are not given. The segments CB and CD represent the lengths of the segments of the secant line CA. So, we can write the equation: AC * BC = CD * CE Substituting the given values: 12 * 3 = CD * CE 36 = CD * CE

Step 3: Identifying that CE is tangent

Notice that CE is tangent to the circle at point E. Therefore, CD and CE are actually the same segment. Then CD = CE. Using this fact, we can rewrite the equation from step 2 as: 36 = CE * CE = CE²

Step 4: Solving for CE

Taking the square root of both sides of the equation CE² = 36 gives CE = 6.