Question

A parallelogram has sides with lengths of `16` and `15`. If the parallelogram's area is `64`, what is the length of its longest diagonal?

Answer 1

The longest diagonal `~~ 30.7`

Explanation:

Here is a reference to the properties of a Parallelogram

Let a = the length of the first side = 15

Let b = the length of the base = 16

The area of a parallelogram is:

`A = bh`

Substitute 64 for the area and 16 for the base:

`64 = 16h`

h = 4

We can use the equation `h = asin(pi - theta)` to find the sine of the angle between the base and the other side.

`4 = 15sin(theta)`

`sin(theta) = 4/15`

Use a well known trigonometric idenity to find the cosine:

`cos(theta) = sqrt(1 - sin^2(theta))`

`cos(theta) = sqrt(1 - (4/15)^2)`

`cos(theta) = sqrt(209)/15`

Because the two angles must add to `pi`, the cosine of the other angle is:

`cos(pi - theta) = cos(pi)cos(theta) + sin(pi)sin(theta)`

Simplify using `cos(pi) = -1 and sin(pi) = 0`

`cos(pi - theta) = -sqrt(209)/15`

The longest diagonal, c, can be found using the law of cosines

`c^2 = a^2 + b^2 - 2abcos(pi - theta)`

`c^2 = 15^2 + 16^2 - 2(16)(15)(-sqrt(209)/15)`

`c^2 = 225 + 256 + 32sqrt(209)`

`c ~~ 30.7`