Question

A parallelogram has sides A, B, C, and D. Sides A and B have a length of `2` and sides C and D have a length of `9`. If the angle between sides A and C is `pi/4`, what is the area of the parallelogram?

Answer 1

The area is `9sqrt2`.

Explanation:

If the side with length 9 (let's say side A) is the base, we need to find the height of the parallelogram. If a line is drawn from the vertex of one of the obtuse angles perpendicular to the base, we would get a right triangle with hypotenuse of length 2. One of its acute angles is `π/4`, and `tan(π/4) = 1`. This means that the two sides other than the hypotenuse have equal lengths.

Using the pythagorean theorem, we can see that, if K, L, M the sides of the right triangle (M is the hypotenuse, and K = L),

`K^2 + L^2 = M^2 => 2L^2 = 4 => L = K = sqrt2`.

This means, since the height of the parallelogram was one of the sides of that right triangle, that the area of the parallelogram is base times height, or `9sqrt2`.