Question Video: Calculating a Rhombus’s Diagonal Length Using Its Area | Nagwa Question Video: Calculating a Rhombus’s Diagonal Length Using Its Area | Nagwa

# Question Video: Calculating a Rhombus’s Diagonal Length Using Its Area Mathematics • Second Year of Preparatory School

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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?

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### 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.

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