Video Transcript
A rhombus has the same area as a square with a four-centimeter diagonal. If one of the diagonals is two centimeters in length, find the length of the other
diagonal.
We’re told that a square and a rhombus are equal in area. Remember, squares and rhombuses are both quadrilaterals with four equal sides. In a square, all the interior angles are right angles, whilst this isn’t the case for
rhombuses in general. We’re told that the diagonal of the square, which is the line segment connecting
opposite vertices, is four centimeters long. One diagonal of the rhombus is two centimeters long, and the length of the other
diagonal is what we’re required to calculate. We’ll label the length of the second diagonal of the rhombus as 𝑑 two
centimeters.
As we’re told that these two quadrilaterals have the same area, we need to recall how
to calculate the area of each. The area of a rhombus is equal to half the product of the lengths of its
diagonals. We can express this as area of rhombus equals a half 𝑑 one multiplied by 𝑑 two. We can’t work this out because we don’t know the value of d two, so let’s consider
the square. The area of a square is, of course, equal to its side length squared. But a square is also a special type of rhombus, in which the two diagonals are of
equal length. So, using the letter 𝑑 to represent the length of each diagonal of the square, we
have the formula area of square equals a half 𝑑 squared.
We know the diagonal of this square is four centimeters, so we can substitute this
value to give that the area of the square is equal to a half multiplied by four
squared. That’s a half multiplied by 16, which is eight, and the units for this area are
square centimeters.
We now know that the area of the rhombus is also eight square centimeters. We can substitute this value and the value two for the length of the first of the
rhombus’s diagonals to give an equation we can solve to find the length of the
second diagonal. That equation is eight equals a half multiplied by two multiplied by 𝑑 two. And remember, 𝑑 two represents the length of the second diagonal. One-half multiplied by two is simply one, so the equation simplifies to eight equals
𝑑 two. We’ve therefore found that the length of the other diagonal of the rhombus is eight
centimeters.
