James says that he can draw several different trapezoids with a height of two and an
area of 29. Charlotte disagrees and said that there is only one trapezoid with height two and
area 29. Who is correct?
Let’s begin by recalling that a trapezoid is a quadrilateral that has one pair of
parallel sides. Now, we need to consider if there is one trapezoid with the given height and area or
more than one. Remember that to find the area of a trapezoid, we calculate one-half times 𝑎 plus 𝑏
times ℎ, where 𝑎 and 𝑏 are the lengths of the parallel bases and ℎ is the
perpendicular height. So let’s consider a trapezoid with a perpendicular height of two length units, which
we could sketch like this.
Now let’s see how we could designate the lengths such that the area is 29 square
units. We can take the general formula for the area and fill in an area of 29 square units
and a height of two length units, which gives us 29 equals one-half times 𝑎 plus 𝑏
times two. Canceling the factor of two from the denominator of one-half and the two being
multiplied would give us 29 equals 𝑎 plus 𝑏.
Remember that 𝑎 plus 𝑏 means the length of the two parallel sides added. So what this means is that these two sides must sum to 29 length units. So we could, for example, create bases of lengths 14 and 15 length units. These two lengths would add to 29 and so the area would be 29 square units.
However, are there other lengths that also sum to 29 length units? Yes, for example, we could draw a trapezoid that has a top base of one unit and a
lower base of 28 length units. These lengths also sum to 29 length units. So the area of this trapezoid would also be 29 square units. In fact, here are two different trapezoids we could also draw. We haven’t even included any trapezoids that have decimal values for the lengths. If we consider all the possible decimal values, then there would be an infinite
number of trapezoids we could draw that have a height of two length units and an
area of 29 square units.
Charlotte said that there was only one trapezoid that we could draw with these
properties. But we can give the answer that James is correct because there is more than one
trapezoid with a height of two and an area of 29.