Find the lengths a, b, c, and d in the following figure:02510502911.png

Find the lengths a, b, c, and d in the following figure:02510502911.png

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asked 2021-04-30

Find the lengths a, b, c, and d in the following figure:
image

Answers (1)

2021-05-02
First find PQ and QR. For PQ, use the right-triangle:
\(\displaystyle{\tan{{\left({24}^{\circ}\right)}}}={\frac{{{1}}}{{{P}{Q}}}}\), so \(\displaystyle{P}{Q}={\frac{{{1}}}{{{\tan{{\left({24}^{\circ}\right\rbrace}}}}}}\)
For QR, first find its opposite angle to be \(\displaystyle{180}^{\circ}-{24}^{\circ}-{48}^{\circ}={108}^{\circ}\)
Now using the law of sines:
\(\displaystyle{\frac{{{\sin{{\left({108}^{\circ}\right)}}}}}{{{Q}{R}}}}={\frac{{{\sin{{\left({24}^{\circ}\right)}}}}}{{{2}}}}\), thus \(\displaystyle{Q}{R}={\frac{{{2}{\sin{{\left({108}^{\circ}\right)}}}}}{{{\sin{{\left({24}^{\circ}\right)}}}}}}\)
Therefore:
\(\displaystyle{P}{R}={P}{Q}+{Q}{R}={\frac{{{1}}}{{{\tan{{\left({24}^{\circ}\right)}}}}}}+{\frac{{{2}{\sin{{\left({108}^{\circ}\right)}}}}}{{{\sin{{\left({24}^{\circ}\right)}}}}}}\approx{6.92}\)
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