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Home » Area of Geometric Shapes » Area of a Trapezoid with Median

Area of a Trapezoid with Median

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In addition to the standard formula for the area of a trapezoid using its bases, we can also calculate the area of a trapezoid with its median and its height. The median is the line connecting the two midpoints of the trapezoid's legs - the non-parallel sides of a trapezoid. The median is also called the midsegment or midline.

Problem

BCD is a trapezoid, AB||CD. EF is a line connecting the midpoints of legs AD and BC, AE=ED and BF=FC. h is the height of the trapezoid. Find a formula for its area using h and |EF|

Area of trapezoid with median

Strategy

Let's see how we can relate what we know about the trapezoid's median to the formula we already have for the area of trapezoid. The area of a trapezoid is (short base+long base)·height/2, or A=½(AB+DC)·h.

In this problem, we have the height, and the median or midsegment. From the trapezoid midsegment theorem, we have the relationship between the midsegment and the bases: |EF|=½(AB+DC). Looking at the two formulas, we see we can simply substitute EF for ½(AB+DC) in the formula for the area and get A=|EF|·h

Solution

(1) A=½(AB+DC)·h //Area of trapezoid
(2) AE=ED, BF=FC //given
(3) EF is the midsegment //(2), Defintion of midsegment
(4) |EF| = ½(AB+DC) //(3), trapezoid midsegment theorem
(5) A = |EF| ·h //(1), (4), substitution

Another way to solve this problem

In the above section, we relied on recognizing that the formula for the area of a trapezoid -A=½(AB+DC)·h looks very similar to the formula for the length of the midsegment - |EF| = ½(AB+DC) , and made a substitution that resulted in a very compact, elegant solution.

But what if we don't immediately recognize that the formulas are similar, or don't remember that the midsegment is equal to half the sum of the bases? Let's look at another way to solve this, without relying on that.

Because EF is a line that connects midpoints of sides, we might be able to use the triangle midsegment theorem - but for that, we need a triangle. So let's draw one, using the diagonal AC:

trapezpid with diagonal

Solution using the triangle midsegment theorem

In triangle ΔACD, |EG| is a line parallel to the base CD, which starts from the midpoint of side AD - so by the converse triangle midsegment theorem, it is a midsegment, and equal to half the base. Let's set |EG|=x. If x is equal to half the base, then the base CD must equal 2x.

Now let's look at the other triangle, ΔACB. Using the same reasoning as above, |GF| starts from the midpoint of side BC and is parallel to AB - so by the converse triangle midsegment theorem, it is a midsegment, and equal to half the base. The length of |GF| is |EF|-x, so base AB is 2·(|EF|-x), or 2·|EF|-2x.

Now let's plug those values into the formula for the area of a trapezoid:

(1) A=½(AB+CD)·h
(2) AB= 2·|EF|-2x
(3) CD=2x
(4) A=½(2·|EF|-2x+2x)·h = ½(2·|EF|)·h = |EF|·h

« Quadrilateral Whose Diagonals Bisect Each Other
Converse of the Trapezoid Midsegment Theorem »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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