# How do you find the missing sides of a triangle given sides are x and 6 root 3, angles are 60 and 90?

Feb 25, 2018

Missing sides and angles are

color(indigo )(x = a = 9, c = 5.1962, hat C = 30^@

#### Explanation:

Given : $a = x , b = 6 \sqrt{3} = 10.3923 , \hat{A} = {60}^{\circ} , \hat{B} = {90}^{\circ}$

Applying law of sines,

$\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C$

$a = x = \frac{b \sin A}{\sin} B = \frac{6 \sqrt{3} \cdot \sin \left(60\right)}{\sin} \left(90\right)$

$a = \frac{6 \sqrt{3} \cdot \sqrt{3}}{2} = 9$

Third angle $\hat{C} = 180 - 90 - 60 = {30}^{\circ}$

As the angles of the right triangle are $60 : 90 : 30$, sides will be in the ratio $\sqrt{3} : 2 : 1$

Third side $c = \left(\frac{1}{2}\right) b = \left(\frac{1}{2}\right) \cdot 6 \sqrt{3} = 5.1962$