Question

In the right triangle shown, \angle B = 60^\circ∠B=60

∘
angle, B, equals, 60, degrees and AB = 12AB=12A, B, equals, 12.
How long is ACACA, C?
Answer exactly, using a radical if needed.

Answer 1

Part 1)
`AC=6√(3)\ units`

Part 2)
`AC=12√(3)\ units`

Explanation:

I will analyze two problems

see the attached figure to better understand the problem

Problem 1

The hypotenuse is the segment AB and the right angle is C

we know that

In the right triangle ABC

`sin(B)=(AC)/(AB)` ---> by SOH (opposite side divided by the hypotenuse)

substitute the given values

`sin(60^o)=(AC)/(12)`

Remember that

`sin(60^o)=(√(3))/(2)`

substitute

`(√(3))/(2)=(AC)/(12)`

`AC=6√(3)\ units`

Problem 2

The hypotenuse is the segment BC and the right angle is A

we know that

In the right triangle ABC

`tan(B)=(AC)/(AB)` ---> by TOA (opposite side divided by the adjacent side)

substitute the given values

`tan(60^o)=(AC)/(12)`

Remember that

`tan(60^o)=√(3)`

substitute

`√(3)=(AC)/(12)`

`AC=12√(3)\ units`