Question

A parallelogram has sides A, B, C, and D. Sides A and B have a length of `18` 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

Use a right triangle to find the height and eventually find the Area = `81sqrt(2)`

Explanation:

We start with a parallelogram with sides 18 and 9 and an angle between the sloping sides is `(pi/4)`. We're asked for the area of the figure.

The area of a parallelogram is:

`A=bh` where b is the base, or long side, and h is the perpendicular distance between the two long sides. To find it, we'll set up a triangle where a part of the base is one side, the height is the other side, and the sloping line (side C or D) is the hypotenuse.

There are 2 angles we know - the intersection of the height with the base, which is a right angle, and the angle between side C and the base, `(pi/4)`, or 45 degrees.

`(pi/4)` is a special angle in geometry - the relationship of the sides of a right triangle that has an angle of `(pi/4)` is `x, x, xsqrt(2)` where the first 2 x terms are the two sides and the `xsqrt(2)` is the hypotenuse.

We can, therefore, divide 9 by the `sqrt(2)` to find the height:

Height = `9/sqrt(2)`

We can multiply this by the long side (side A = 18) to arrive at:

`A=bh`

`A=18*9/sqrt(2)`

Let's clean this up by getting rid of the square root in the denominator and working the math:

`A=(18*9)/sqrt(2)*sqrt(2)/sqrt(2)=((18)(9)sqrt(2))/2=81sqrt(2)`