Video Transcript
In the rhombus 𝐴𝐵𝐶𝐷, the side
length is 8.5 centimeters, and the diagonal lengths are 13 centimeters and 11
centimeters. Find the length of line segment
𝐷𝐹. Round your answer to the nearest
10th.
We can begin this question by
recognizing that a rhombus is a quadrilateral that has all four sides of the same
length. We’re told that this side length is
8.5 centimeters, so we can label this on the diagram. We can also label the two
diagonals. One of them is 13 centimeters, and
one is 11 centimeters. It’s always nice to see if we can
get these in the correct positions. And as the diagonal 𝐴𝐶 looks
longer than the length of 𝐵𝐷, then it will be 13 centimeters. We’re asked to find the length of
this line segment, 𝐷𝐹. If we look at the diagram, we
should notice that this length of 𝐷𝐹 is in fact the perpendicular height of the
rhombus. So, how could we link the diagonals
of the rhombus with the perpendicular height? Well, we can in fact do this using
the formulas for the area of a rhombus.
The first formula, we should
remember, is that the area of a rhombus is calculated by multiplying the two
diagonals 𝑑 sub one and 𝑑 sub two and then halving it. The second formula tells us that
the area of a rhombus is equal to the base multiplied by the perpendicular
height. As we’re given the lengths of the
diagonals in this question, let’s fill these values in to our first formula. We therefore calculate 11
multiplied by 13 divided by two. As we’re asked for our answers to
the nearest 10th, we can assume that we’re allowed to use a calculator. So, we can give our answer as 71.5
square centimeters.
Now, we found the area of a
rhombus, we can plug our value in to the second formula. On the left-hand side, we’ll have
the area as 71.5. The base will be the length of the
rhombus, which is 8.5 centimeters. And we’re trying to work out the
unknown perpendicular height, which we can leave as ℎ. In order to find the value of ℎ, we
would divide both sides of our equation by 8.5, which gives us 8.41176 and so on is
equal to ℎ. As we need to round our answer to
the nearest 10th, we would check our second decimal digit to see if it’s five or
more. And as it isn’t, then our value of
ℎ would round to 8.4 centimeters. We know that the length of line
segment 𝐷𝐹 is the same as the perpendicular height of the rhombus. So, our answer is that the length
of line segment 𝐷𝐹 is 8.4 centimeters.
