Find the measure of the angle!

Geometry Level pending

Let $C_{r}$ and $C_{b}$ represent the red and the blue circles respectively.

$C_{r}$ and $C_{b}$ have radii $r$ and $R$ respectively, where $R > r$ , and $P$ is a point on $C_{b}$ and both circles are tangent to each other at $A$ and $\overline{PM} = \sqrt{R(R - r)}$ is tangent to $C_{r}$ at $M$ and $O'$ and $O$ are the centers of $C_{r}$ and $C_{b}$ respectively.

Find the measure of the acute $\angle{PAD}$ in degrees.

The answer is 60.

1 solution

Rocco Dalto
Apr 29, 2021

$\overline{OP} = \overline{OA} = R \implies m\angle{PAD} = m\angle{APO} = \theta \implies m\angle{POA} = 180^{\circ} - 2\theta$

Using the Pythagorean Theorem on right $\triangle{O'PM} \implies \overline{O'P} = \sqrt{r^2 +R^2 - Rr}$

Using the law of cosines on isosceles $\triangle{APO}$ with included $\angle{POA} \implies$

$\overline{AP^2} = 2R^2(1 + \cos(2\theta)) = 4R^2\cos^2(\theta) \implies \overline{AP} = 2R\cos(\theta)$

Using the law of cosines on $\triangle{APO'}$ with included $\angle{PAD} \implies$

$r^2 + R^2 - Rr = 4R^2\cos^2(\theta) + r^2 - 4Rr\cos^2(\theta) \implies$

$R(R - r) = 4R(R - r)\cos^2(\theta) \implies \cos^2(\theta) = \dfrac{1}{4} \implies \cos(\theta) = \dfrac{1}{2} \implies \boxed{\theta = 60^{\circ}}$ .

Find the measure of the angle!

The answer is 60.

1 solution

0 pending reports