Question

AC = 12√3. Find BC and AB. Write answer in simplest form.

Answer 1

BC = a

AC = b= 12√3

AB =c

A= 30°

B=60°

C=90°

Using the sine rule

`\frac{\sin\text{ A}}{a}=(\sin B)/(b)`

substitute the values into the above

`(\sin30)/(a)=\frac{\sin 60}{12\sqrt[]{3}}`
`((1)/(2))/(a)=\frac{\frac{\sqrt[]{3}}{2}}{12\sqrt[]{3}}`
`(1)/(2* a)=\frac{\sqrt[]{3}}{2*12\sqrt[]{3}}`
`(1)/(2a)=\frac{\sqrt[]{3}}{24\sqrt[]{3}}`
`(1)/(2a)=(1)/(24)`

cross multiply

`2a=\text{ 24}`
`a=12`

Therefore BC = 12

Let's proceed to find AB

`(\sin A)/(a)=(\sin C)/(c)`
`(\sin30)/(12)=(\sin 90)/(c)`
`((1)/(2))/(12)=(1)/(c)`
`(1)/(2*12)=(1)/(c)`
`(1)/(24)=(1)/(c)`

cross-multiply

`c=24`