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# Even More Trig (I'm Sorry)

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Right triangle ABC is inscribed in equilateral triangle PQR, as shown. Given PB = 2, PC = 5, and QC = 3, then find QA.

Diagram: https://latex.artofproblemsolving.com/miscpdf/utullifn.pdf?t=1577916255464.

I am just adding the pic.  (Melody).

Jan 1, 2020
edited by Melody  Jan 1, 2020

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I may not have done this the best way but this was my method:

1)  Find BC

2)  find angle BCP

3) Find angle ACQ

4) Find angfle CAQ

5) Now you have 3 angles and one side of triangle ACQ so you can find AQ

Jan 1, 2020
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1) How would I find BC?

MathCuber  Jan 2, 2020
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You can find BC using cosine rule.  You have 2 sides and the included angle so you can find the 3rd side.

Melody  Jan 2, 2020
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So BC is $$\sqrt{19}$$.

Then, by the law of sines, sin BCP is $$\frac{\sqrt3}{\sqrt{19}}$$.

What now?

MathCuber  Jan 2, 2020
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If

$$sin\angle{PCB}=\frac{\sqrt3}{\sqrt{19}}\\ then\\ \angle{PCB}=sin^{-1}\frac{\sqrt3}{\sqrt{19}}\\ \angle{PCB}\approx 23.413 ^{\circ}$$

Now, check you know how I did that on a calculator and then move onto step 3

sorry it has not displayed properly, I will see if I can fix it.    It is fixed I hope.

Melody  Jan 2, 2020
edited by Melody  Jan 2, 2020
edited by Melody  Jan 2, 2020
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Thanks for the help! I figured out the answer! :)

MathCuber  Jan 2, 2020