Question

A parallelogram has sides with lengths of `15` and `8`. If the parallelogram's area is `8`, what is the length of its longest diagonal?

Answer 1

Longest Diagonal: `color(magenta)(22.99)` (approx.)

Explanation:

The height of the parallelogram, `color(blue)(h)`, relative to the side with length `15` is
`color(white)("XXX")color(blue)(h)=color(red)("Area")/15 = color(red)(8)/15`

Extending the side with length `15` to form a right triangle as in the image below:

Applying the Pythagorean Theorem:

We can see that the length of the extension, `color(green)(x)` is
`color(white)("XXX")color(green)(x)=sqrt(8^2-color(blue)(h^2))`

`color(white)("XXXX")=sqrt(8^2-(8/15)^2) ~~7.892`

And the longest diagonal of the original parallelogram is
`color(white)("XXX")sqrt((15+color(green)(x))^2+color(blue)(h)^2)`

`color(white)("XXX")~~22.99`