# A triangle has sides A, B, and C. Sides A and B have lengths of 5 and 9, respectively. The angle between A and C is (17pi)/24 and the angle between B and C is  (pi)/8. What is the area of the triangle?

Jan 19, 2016

area of the triangle$\approx 12.889 \text{ square unites}$ to 3 decimal places

#### Explanation:

$\textcolor{b l u e}{\text{Method}}$
Find the height h then use: "area=1/2xxAxxh
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$\textcolor{b l u e}{\text{Consider the option to find C and hence h}}$

Cosine Rule:
${C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos \left(\angle b c a\right)$

Sine Rul:
$\frac{C}{\sin \left(\angle b c a\right)} = \frac{B}{\sin \left(\angle a b c\right)} = \frac{A}{\sin \left(\angle b a c\right)}$

Both require that the angle $\angle b c a$ is determined.
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Using: Sum of internal angles of a triangle$= {180}^{o} \to \left(\pi \text{ radians}\right)$

$\textcolor{b l u e}{\implies \angle b c a = \pi - \frac{17 \pi}{24} - \frac{\pi}{8} = \frac{\pi}{6} \to \left({30}^{o}\right)}$

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$\textcolor{b l u e}{\text{Determining length of C}}$

Using the Sine Rule (simpler: no square roots!)

$\frac{C}{\sin \left(\frac{\pi}{6}\right)} = \frac{A}{\sin \left(\frac{\pi}{8}\right)}$

Known: $\textcolor{w h i t e}{\ldots} \sin \left(\frac{\pi}{6}\right) = \sin \left({30}^{o}\right) = \frac{1}{2}$

$\textcolor{b r o w n}{\implies C = \frac{1}{2} \times \frac{5}{\sin \left(\frac{\pi}{8}\right)} \approx 6.499}$ to 3 decimal places
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$\textcolor{b l u e}{\text{Determining height h}}$

Project the line cb to d such that a vertical line from 'a' forms 'ad' and $\angle b \mathrm{da} = \frac{\pi}{2} \to \left({90}^{o}\right)$

Then $\angle \mathrm{db} a = \pi - \frac{17 \pi}{24} = \frac{7 \pi}{24} \to 52 \frac{1}{2} \text{ degrees}$

Using basic trig
$\sin \left(\frac{7 \pi}{24}\right) = \frac{h}{C}$

$\implies h = C \sin \left(\frac{7 \pi}{24}\right)$

$\textcolor{b r o w n}{h \approx 5.156}$ to 3 decimal places
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$\textcolor{b l u e}{\text{To determine area of triangle}}$

using: area $= \frac{1}{2} \times A \times h$

color(brown)("area "= 1/2xx 5xx5.156... ~~ 12.889" square unites" to 3 decimal places