Question

If sides A and B of a triangle have lengths of 2 and 4 respectively, and the angle between them is `(pi)/4`, then what is the area of the triangle?

Answer 1

`2sqrt(2)`

Explanation:

One way to do this is strictly geometrical.

Consider the square containing the given triangle with the side of length `4` as the diagonal.
We would have the structure below:

We are asked for the area of `triangleT`
and it is clearly
`color(white)("XXX")`the area of the `square`
`color(white)("XXX")`minus
`color(white)("XXX")`(the area of `triangleA` plus `triangleB`)

With a diagonal of length `4` each side of the square has a length of `2sqrt(2)` (as shown) based on the Pythagorean Theorem.
Therefore the area of the `square` is `2sqrt(2) xx 2sqrt(2) = 8`

The area of `triangleA` is (`1/2bh`)
`color(white)("XXX")1/2xx2sqrt(2)xx2= 4`

The area of `triangleB` is
`color(white)("XXX")1/2xx(2sqrt(2)-2)xx2sqrt(2) = 4-2sqrt(2)`

The area of `triangleT` is
`color(white)("XXX")8- (4+4-2sqrt(2)) = 2sqrt(2)`