# How do you solve the triangle given a=8, b=24, c=18?

Feb 24, 2018

Three angles are color(blue)(hat A = 14.63^@, hat B = 130.75^@, hat C = 34.62^@

#### Explanation:

Known three sides. To find the three angles.

${A}_{t} = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$ where s is the semi perimeter of the triangle.

$s = \frac{8 + 24 + 18}{2} = 25$

${A}_{t} = \sqrt{25 \cdot \left(25 - 8\right) \left(25 - 24\right) \left(25 - 18\right)} = 54.54$

But ${A}_{t} = \left(\frac{1}{2}\right) b c \sin A$

$\therefore \sin A = \frac{2 \cdot {A}_{t}}{b c} = \frac{2 \cdot 54.54}{24 \cdot 18} = 0.2525$

$\hat{A} = {\sin}^{-} 1 0.2525 = {14.63}^{\circ}$

Similarly,
$\hat{B} = {\sin}^{-} 1 \left(\frac{2 \cdot 54.54}{8 \cdot 18}\right) = \left(180 - 49.24\right) = {130.75}^{\circ}$ since it is an obtuse angle.

$\hat{C} = {\sin}^{-} 1 \left(\frac{2 \cdot 54.54}{8 \cdot 24}\right) = 34.62$