Dealing with Circles

Geometry Level 4

$AB$ is a diameter of a circle of radius $10$ . $C$ is a point on the circle such that arc $BC=\dfrac{5\pi}{3}$ . The bisector of $\angle ACB$ cuts the circle at $D$ . Find the length of $CD$ .

Give your answer to 2 decimal places.

The answer is 17.32.

1 solution

Ayush G Rai
May 24, 2016

Let $O$ be the center of the circle.Then $\angle BOC=\frac{5\pi}{3}\times \frac{1}{10}=\frac{\pi}{6}.$ Since $CD$ bisects $\angle CB,D$ is the midpoint of the arc $ADB.$ Therefore $OD\perp AB.$
Hence $\angle COD=\frac{\pi}{6}+\frac{\pi}{2}=\frac{2\pi}{3}$ and corresponding $CD=2\times 10\times sin\frac{\pi}{3}=10\sqrt3$ ~ $\boxed {17.32}$

Ah, I thought the question was for the length of the arc CD. Maybe ask for "length of chord CD"? Nonetheless, great problem!

G Silb - 1 year, 11 months ago

I thought that the word length would indicate that is a straight line

Ayush G Rai - 1 year, 9 months ago

Dealing with Circles

The answer is 17.32.

1 solution

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