Question

$\overline{ab}$ is a radius. $\overleftrightarrow{cd}$ is tangent to circle $a$ at point $b$, where $ec = 32$ and $ab = 9$.
what is $bc$?
enter your answer in the box.
$bc = \square$

Explanation:

Step1: Define variables and use tangent rule

Let $BC = x$, $AC = AE + EC = 9 + 32 = 41$. Since tangent $CD$ is perpendicular to radius $AB$, $\triangle ABC$ is right-angled at $B$. By Pythagoras theorem:
$$AB^2 + BC^2 = AC^2$$

Step2: Substitute values and solve for $x$

Substitute $AB=9$, $AC=41$, $BC=x$:
$$9^2 + x^2 = 41^2$$
$$81 + x^2 = 1681$$
$$x^2 = 1681 - 81 = 1600$$
$$x = \sqrt{1600} = 40$$

Answer:

$BC = 40$