Question

A parallelogram has sides with lengths of `5` and `9`. If the parallelogram's area is `45`, what is the length of its longest diagonal?

Answer 1

Diagonal length: `sqrt(106)~~10.3` units

Explanation:

With sides of `5` and `9`
and and area of `45` sq. units
the parallelogram is actually a rectangle.

`color(white)("XXX")`To see this, notice that using `9` as the base
`color(white)("XXX")`with an area of `45`, the perpendicular height
`color(white)("XXX")`must be `5` units.

The length of the diagonal of a rectangle can be calculated based on the length of its sides, using the Pythagorean Theorem.
In this case:
`color(white)("XXX")"diagonal length "=sqrt(5^2+9^2)`

`color(white)("XXXXXXXXXXXXX")=sqrt(25+81)`

`color(white)("XXXXXXXXXXXXX")=sqrt(106)`

Answer 2

I am advised that the found condition is mathematically compatible with the shape being a rectangle.

`L=sqrt(106) larr` Exact answer

`L.~~10.296` to 3 decimal places`larr` Approximate answer

Explanation:

When in doubt do a very quick and rough sketch in the margin. For this question it should take about 5 to 6 seconds

`color(blue)("Comment")`
Notice that `5xx9=45` which is the same as the given area. Thus the parallelogram is also a rectangle. Thus you can go straight in and use Pythagoras. `L=sqrt(5^2+9^2)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Beginnings of a mathematical approach if angle "GAB<90^o)`

`color(blue)("Method")`
`color(brown)("The longest diagonal is AD")`

Determine the length of DE (h) using area
Determine the length of FE and hence AE
Solve for AD using Pythagoras
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine h")`

If you were to get a pair of scissors and cut out triangle ABG you will discover that it will fit exactly over the empty space DEF. The result being a rectangle.

The area of the rectangle is:
`9" units"xxh" units"->9h = 45" units"^2`

Divide both sides by `color(red)(9)`

`color(green)(9h=45color(white)("ddd")->color(white)("ddd")9/color(red)(9)h=45/color(red)(9) )`

But `9/9=1` giving:

`color(green)(color(white)("ddddddddd")->color(white)("ddddd")h=45/9`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Determine FE")`

Now we find out that `h=45/9=5` which is the same length as the slope. Consequently FE=0 and `color(magenta)("The shape is a rectangle")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Greatest slope length is the diagonal of the rectangle.

Let the slope length be `L`

`L=sqrt(9^2+5^2)= sqrt(106)=sqrt(2xx56)`

Both 2 and 56 are prime numbers so we can not simplify any further

`L=sqrt(106)` Exact answer

`L=10.29563....~~10.296` to 3 decimal places