A geometry problem by Lee Dongheng

Geometry Level 3

In the diagram below, B C BC is the diameter and O D OD is the radius of the semicircle centered at O . O. If A D = D C , AD=DC, what is sin O A C ? \sin \angle OAC?

Give your answer correct to three decimal places.


The answer is 0.316.

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2 solutions

Lee Dongheng
Aug 10, 2017

We can calculate the answer without using calculator until the final answer, here is a way:

Let O E = a OE=a , O E OE is drawn such that O E D C OE \perp DC

O C = 2 a , B O = O C = 2 a \therefore OC=\sqrt{2}a, BO=OC=\sqrt{2}a

A B = 2 2 a \therefore AB=2\sqrt{2}a

A O = ( 2 2 a ) 2 + ( 2 a ) 2 = 10 a \therefore AO= \sqrt{(2\sqrt{2}a)^{2}+(\sqrt{2}a)^{2}}=\sqrt{10}a

s i n O A C = O E A O = a 10 a = 10 10 = 0.316 \therefore sin \angle OAC = \frac{OE}{AO} = \frac{a}{\sqrt{10}a}=\boxed{\frac{\sqrt{10}}{10}=0.316}

Marta Reece
Aug 9, 2017

O D = O C O D C = B A C = 4 5 OD=OC\implies\measuredangle ODC=\measuredangle BAC=45^\circ

A B = B C = 2 × B O AB=BC=2\times BO

B A C = arctan 1 2 26.56 5 \measuredangle BAC=\arctan\frac12\approx26.565^\circ

O A C = 4 5 26.56 5 = 18.43 5 \measuredangle OAC=45^\circ-26.565^\circ=18.435^\circ

sin O A C = sin ( 18.43 5 ) = 0.316 \sin OAC=\sin(18.435^\circ)=\boxed{0.316}

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