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# help

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Find the length of diagonal CD.

Jun 13, 2020

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Because AB is a diameter, by Thale's theorem,

$$\angle ACB = \angle ADB = 90^\circ$$

That means $$\angle CAB = \arccos\left(\dfrac24\right) = 60^\circ$$ by simple trigonometry.

Also, because $$\angle ADB = 90^\circ$$, considering the interior angle sum of $$\triangle ADB$$,

$$\angle DBA = 45^\circ$$. This means $$\triangle ADB$$ is a right isosceles triangle, which means AD = DB.

By Pythagorean theorem, we get $$AD = DB = 2\sqrt 2$$.

Now consider $$\triangle ADC$$ and use Law of Cosine. I believe you can finish it from here on your own.

Jun 13, 2020