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?
The shape that we’re interested in in this question is a rhombus. That’s a quadrilateral, a four-sided shape, which has two pairs of parallel sides. And all four of its sides are of equal length. It looks something like this. The diagonals of a rhombus are the lines which connect opposite corners to one another. So they’re the lines that I’ve marked in pink. We’re told that, in this rhombus, one of the diagonals is twice the length of the other diagonal. So if we call the shorter diagonal 𝑑, we can refer to the longer diagonal as two 𝑑.
Next, we’re told that the area of this rhombus is 81 square millimeters. And we’re asked to work out the lengths of the diagonals. So we need to recall how the area of a rhombus can be calculated. There’re actually two ways to work out the area of a rhombus depending on the information we’ve been given. A rhombus is just a special type of parallelogram. It’s a parallelogram in which all four side lengths are equal. So the first way is to multiply the base by the perpendicular height. Or if we’ve been given the lengths of the diagonals of the rhombus, so 𝑑 one and 𝑑 two, we can find its area by finding a half of 𝑑 one multiplied by 𝑑 two.
In this question, it’s information about the diagonals that we’ve been given. So we’re going to use the second formula for finding an area. We have a half multiplied by 𝑑 one and 𝑑 two, where 𝑑 one and 𝑑 two are the lengths of the rhombus’s diagonals. In our rhombus, we’ve already labelled the lengths of the diagonals as two 𝑑 and 𝑑 using the information we were given early in the question. So we have a half multiplied by two 𝑑 multiplied by 𝑑. This is equal to the value we were given for the area, 81 square millimeters.
Now, we can solve this equation to find the value of 𝑑. First, we can cancel a factor of two. There’s a two in the denominator and a two in the numerator. Or you can think of this as a half multiplied by two, which is equal to one. So our equation simplifies to 𝑑 multiplied by 𝑑 equals 81. 𝑑 multiplied by 𝑑 can be written as 𝑑 squared. To find the value of 𝑑, we need to square root both sides of this equation, giving 𝑑 is equal to the square root of 81, which is just nine.
Normally, when we solve an equation by square rooting, we must remember to take plus or minus the square root. But here 𝑑 represents a length. So it must take a positive value. So we found the length of the shorter diagonal of the rhombus. It’s nine millimeters. To find the length of the longer diagonal, we need to multiply this by two. Nine multiplied by two is 18.
So we found that the lengths of the diagonals of this rhombus are nine millimeters and 18 millimeters.