Question

A parallelogram has sides with lengths of `15` and `12`. If the parallelogram's area is `48`, what is the length of its longest diagonal?

Answer 1

The longer diagonal is `26.76(2dp) unit`

Explanation:

We know the area of the parallelogram as `A_p=s_1*s_2*sin theta or sin theta=48/(15*12)=0.2666 :. theta=sin^-1(0.2666)=15.466^0:.`consecutive angles are supplementary `:.theta_2=180-15.466=164.534^0`.
Longer diagonal can be found by applying cosine law:`d_l= sqrt(s_1^2+s_2^2-2*s_1*s_2*costheta_2)=sqrt(15^2+12^2-2*15*12*cos164.534) =26.76(2dp) unit`[Ans]

Answer 2

Longest diagonal `~~ 26.7575`
(this answer included as a more Geometric solution than the excellent answer by Binayaka C which relies on more Trigonometric knowledge)

Explanation:

We are asked to find the length of the longest diagonal of a parallelogram.
We could, in general, imagine an image like the one below.

For this specific question we are told that
`color(white)("XXX")a=12`;
`color(white)("XXX")b=15`;
`color(white)("XXX")`and, although we won't use it yet, `Area =48`
`color(magenta)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`

One of the Geometric tools that we have for finding a length is the Pythagorean Theorem.
If we could extend the base (the side with length `b`) to form a right-angled triangle, as in the diagram below, we could use the Pythagorean Theorem to find the required diagonal length.

For this specific question, we know that
`color(white)("XXX")color(brown)(h) = color(green)(48)/15=16/5=color(brown)(3.2)`
`color(magenta)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`

We are only left with finding the length of the extended side,`color(orange)(k)` in order to be able to apply the Pythagorean Theorem.

But `color(orange)(k)` is just composed of the original length, `b`, plus an extension of `color(blue)(x)`

And we can find the value of `color(blue)(x)` by applying the Pythagorean Theorem to the smaller right-angled triangle with hypotenuse `a` and the previously determined `color(brown)(h)`

For this specific question
`color(white)("XXX")color(blue)(x) = sqrt(12^2-3.2^2) ~~11.56547` (that's why we have calculators)

The length of the extended base is
`color(white)("XXX")color(orange)(k)=15+11.56547=color(orange)(26.56547)`
By the Pythagorean Theorem, the length of the required diagonal is
`color(white)("XXX")color(red)(d)=sqrt(color(orange)(26.56547)^2+color(brown)(3.2)^2)~~color(red)(26.7575)`