Question

The length $l$ of a tangent, drawn from a point $A$ to a circle, is $\dfrac43$ of the radius $r.$ The (shortest) distance from $A$ to the circle is:

$\dfrac12r$

$r$

$\dfrac12l$

$\dfrac23l$

a value between $r$ and $l$

Answer: C

Concepts:tangent line right triangle circle

Difficulty rating: 1280

Small Hint:

Join $A$ to the center and to the point of tangency

Big Hint:

Use the right triangle with legs $r$ and $l=\frac{4r}{3}$

Solution:

If $O$ is the center and $T$ the point of tangency, then $OT\perp AT.$ Hence $\begin{aligned} AO &=\sqrt{r^2+l^2}\\ &=\sqrt{r^2+\frac{16r^2}{9}}\\ &=\frac{5r}{3}. \end{aligned}$ The shortest distance from $A$ to the circle is $AO-r=\frac{2r}{3}.$ Since $l=\frac{4r}{3},$ this distance equals $\frac{l}{2}.$

Therefore, the correct answer is C.

1959 AMC 12 Problem 32

Answer: C

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 32 in Other Years