Question

$\overline{ab}$ is a radius. $\overleftrightarrow{cd}$ is tangent to circle $a$ at point $b$, where $ec = 4$ and $bc = 8$.
what is $ab$
enter your answer in the box.
$ab = \square$ units

Explanation:

Step1: Define variables for radius

Let $AB = AE = r$ (radii of circle $A$). Then $AC = AE + EC = r + 4$.

Step2: Use tangent-radius theorem

Since $CD$ is tangent to circle $A$ at $B$, $\angle ABC = 90^\circ$, so $\triangle ABC$ is a right triangle. Apply the Pythagorean theorem:
$$AB^2 + BC^2 = AC^2$$

Step3: Substitute known values

Substitute $AB=r$, $BC=8$, $AC=r+4$ into the formula:
$$r^2 + 8^2 = (r + 4)^2$$

Step4: Expand and simplify equation

Expand the right-hand side:
$$r^2 + 64 = r^2 + 8r + 16$$
Cancel $r^2$ from both sides, then solve for $r$:
$$64 = 8r + 16$$
$$8r = 64 - 16 = 48$$
$$r = \frac{48}{8} = 6$$

Answer:

$AB = 6$ units