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Question Video: Determining the Number of Possible Trapezoids with a Given Height and Area Mathematics • 11th Grade

James said that he can draw several different trapezoids with a height of 2 and an area of 29. Charlotte disagrees and said that there is only one trapezoid with height 2 and area 29. Who is correct?

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Video Transcript

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.

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