If the area of a trapezoid is 1377 square centimetres and its height is 51 centimetres, find the length of its middle base.
Let’s start by drawing a diagram of our trapezoid, which will have a pair of parallel sides. We’re told that the area is 1377 square centimetres and that the height is 51 centimetres. Notice that the height here refers to the perpendicular or vertical height and not the length of one of the sides. We’re asked to find the length of the middle base. This is the line joining the midpoints of our nonparallel sides. And it’s often called the midsegment of the trapezoid.
We can recall that the length of the midsegment of a trapezoid is equal to half the sum of the lengths of the two parallel sides. So if we called our parallel lengths 𝑎 and 𝑏, then we can write the length of the midsegment as 𝑎 plus 𝑏 over two. So if we returned to our diagram, we can see that we don’t actually know the lengths of our parallel sides. So let’s see if we can use the area to help us. We’ll need to use another fact to help us. That is, that the area of a trapezoid is equal to 𝑎 plus 𝑏 over two times ℎ, where 𝑎 and 𝑏 are the parallel lengths and ℎ refers to the height of our trapezoid.
So let’s start by writing down the area formula and then filling in the information we’ve been given. Since we have an area of 1377 and a height of 51, we’ll have 1377 equals 𝑎 plus 𝑏 over two times 51. To find 𝑎 plus 𝑏 over two on its own, we need to do the inverse operation to multiplying by 51. And we divide both sides of our equation by 51. We can then calculate that 1377 divided by 51 is 27. So we have 27 equals 𝑎 plus 𝑏 over two.
At this point, it looks like we haven’t got very far. We still don’t know 𝑎 or 𝑏. But if we look again at our facts on the midsegment of a trapezoid, we can see that the midsegment is equal to 𝑎 plus 𝑏 over two, which is exactly what we find in our calculation. Therefore, we can say that our midsegment or middle base length is 27 centimetres.