Question

[Precalculus] Find the area of this triangle:

Answer 1

Look at the picture.

In a 30-60-90 (45-45-90) triangle, you can find the measure of any of the three sides (picture 2).

`CD=(1)/(2)AC\to CD=(1)/(2)\cdot12=6`

`AD=CD\sqrt3\to AD=6\sqrt3`

`DB=CD\to DB=6`

`AB=AD+DB\to AB=6\sqrt3+6`

The formula of the area of a triangle ABC:

`A_\triangle=(1)/(2)(AB)(CD)`

substitute:

`A_\triangle=(1)/(2)\cdot(6\sqrt3+6)(6)=3(6\sqrt3+6)=18+18\sqrt3=18(1+\sqrt3)`

Answer:
`\boxed{C.\ 18(1+\sqrt3)}`

Answer 2

check the picture below.

notice, by drawing the altitude from C, we end up with a 30-60-90 and a 45-45-90 triangle, thus we use the 30-60-90 and 45-45-90 rules as you see there.

`image`