A trapezium is shown below. The length of side 𝑧 is six times the length of side 𝑤. The area of the trapezium is 409.5 square centimeters. Calculate length 𝑧.
So first, we know that it’s a trapezium. And the length of side 𝑧 is six times the length of side 𝑤. So we could say that 𝑧 is equal to six 𝑤. We’re also told the area of the trapezium.
The area of a trapezium is one-half times 𝑎 plus 𝑏 times the height, where 𝑎 and
𝑏 are the bases and in this case, 𝑤 and 𝑧, 𝑎 and 𝑏. And we know what the area actually is: 409.5 square centimeters.
We replace 𝐴 with 409.5, 𝑎 and 𝑏 with 𝑤 and 𝑧, and the height of the trapezium
is 18. Since we’re trying to find the length of 𝑧, we either need to solve for 𝑧 with our
equation or solve for 𝑤.
And then, once we know 𝑤, substitute it in here, and take that number times six. And that will be the length of 𝑧. So let’s go ahead and replace 𝑧 with six 𝑤 and now simplify.
𝑤 plus six 𝑤 is seven 𝑤. We can also take one-half times 18, which gives us nine. And if we take nine times seven 𝑤 on the right-hand side of the equation, we get
409.5 is equal to 63𝑤.
Dividing both sides by 63, we find that 𝑤 is equal to 6.5. And we’re to find the length of 𝑧. And since 𝑧 is equal to six 𝑤, we need to take six times six and a half, resulting
Therefore, the length of 𝑧 will be 39 centimeters.