### Video Transcript

One diagonal of a rhombus is twice
the length of the other diagonal. If the area of the rhombus is 81
square millimeters, what are the lengths of the diagonals?

In this question, we’re required to
relate the area of a rhombus to the lengths of its diagonals. So we recall that the area of a
rhombus is equal to half the product of the lengths of its diagonals. It’s equal to 𝑑 one 𝑑 two over
two, where 𝑑 one and 𝑑 two are the lengths of the diagonals. We’ve been given that the area of
this rhombus is 81 square millimeters, but not the lengths of either diagonal. However, we do know that one
diagonal is twice the length of the other. So we could let diagonal length 𝑑
one be equal to twice the length of diagonal 𝑑 two.

We can then form an equation. Replacing 𝑑 one with its
expression in terms of 𝑑 two and the area of the rhombus with 81, we have the
equation two 𝑑 two multiplied by 𝑑 two over two is equal to 81. The twos in the numerator and
denominator of this fraction will cancel each other out. And we’re left with 𝑑 two
multiplied by 𝑑 two, or 𝑑 two squared, is equal to 81. To find the value of 𝑑 two, we
take the square root of both sides of this equation, taking only the positive value
as 𝑑 two represents a length. So 𝑑 two is equal to the positive
square root of 81, which is nine.

So we found the length of one
diagonal. And to find the length of the
other, we need to double this value. 𝑑 one is two times nine, which is
18. So, by recalling that the area of a
rhombus is half the product of the lengths of its diagonals, which we used to set up
and then solve an equation, we found that the lengths of the diagonals of this
rhombus are nine millimeters and 18 millimeters.