Question

Find the missing side lengths. Leave your answers as radicals in simplest form. I need help quickly!

Answer 1

`m = (4)/(√(3)) \text{ or, in rational form: } m = (4√(3))/(3)`

`n = (2)/(√(3)) \text{ or, in rational form: } n = (2√(3))/(3)`

Not sure which form your teacher wants the answers, would suggest putting in both

Explanation:

The missing angle of the triangle = 180 - (60 + 90) = 30°

We will use the law of sines to find m and n

The law of sines states that the ratio of each side to the sine of the opposite angle is the same for all sides and angles

Therefore since m is the side opposite 90° and 2 is the side opposite 60°,

`(m)/(\sin 90) = (2)/(\sin 60)}\\\\`

sin 90 = 1

sin 60 = √3/2

So

`(m)/(1) = (2)/(√(3)/2) \\\\m = (2)/(√(3)/2) \\\\m = (2 \cdot 2)/(√(3)) \\\\m = (4)/(√(3))\\\\`

We can rationalize the denominator by multiplying numerator and denominator by √3 to get

`m = (4√(3))/(3)`
(I am not sure what your teacher wants, you can put both expressions, they are the same)

To find n
Using the law of sines we get

`(n)/(\sin 30) = (m)/(\sin 90)\\\\(n)/(\sin 30) = m\\\\(n)/(\sin 30) = (4)/(√(3))\\\\`

sin 30 = 1/2 giving

`(n)/(1/2) = (4)/(√(3))\\\\n = (1/2 \cdot 4)/(√(3)) \\\\n = (2)/(√(3))`

In rationalized form

`n = (2√(3))/(3)}`