Question

line dc is tangent to circle a. what is the length of $overline{bd}$? o a) 6 in. o b) 9 in. o c) 12 in. o d) 15 in.

Explanation:

Step1: Recall tangent - radius property

Since line $DC$ is tangent to circle $A$ at point $C$, then $\angle ACD = 90^{\circ}$. And $AC = AB=9$ inches (radii of the same circle). Let $BD=x$. Then $AD=AB + BD=9 + x$. In right - triangle $ACD$, $AC = 9$ inches and $CD = 12$ inches.

Step2: Apply Pythagorean theorem

By the Pythagorean theorem in right - triangle $ACD$, $AC^{2}+CD^{2}=AD^{2}$. Substitute $AC = 9$, $CD = 12$ into the formula: $9^{2}+12^{2}=(9 + x)^{2}$. Calculate $9^{2}=81$ and $12^{2}=144$, so $81 + 144=(9 + x)^{2}$, which simplifies to $225=(9 + x)^{2}$.

Step3: Solve for $x$

Take the square root of both sides: $\sqrt{225}=9 + x$. Since $\sqrt{225}=15$, we have $15=9 + x$. Then $x=15 - 9=6$.

Answer:

A. 6 in.