The Trapezoid Midsegment (Median) Theorem

The midsegment of a trapezoid, also called its median, is the segment connecting the midpoints of the two legs. It runs across the middle of the figure, parallel to both bases, and its length has a satisfying answer: it is the average of the two bases.

That makes this one of the shortest rules in geometry. Halfway up the trapezoid, halfway between the two lengths. Every problem in this lesson is that one formula, used forward to find the midsegment or backward to find a missing base.

The formula

The midsegment connects the midpoints of the legs, which are the two non-parallel sides. Do not connect the midpoints of the bases; that segment is something else entirely. Once drawn, the midsegment is parallel to both bases.

Its length is $m = \dfrac{b_1 + b_2}{2}$ , where $b_1$ and $b_2$ are the two bases. Because it is an average, the midsegment always lands between the two base lengths. If your answer is bigger than the long base or smaller than the short base, you added when you should have averaged.

Running the formula backward

Problems often give you the midsegment and one base and ask for the other. Substitute what you know and solve. From $m = \dfrac{b_1 + b_2}{2}$ , multiplying both sides by $2$ gives $2m = b_1 + b_2$ , so twice the midsegment equals the two bases added together.

That doubled form is the fastest way through these: double the midsegment, then subtract the base you were given. If the midsegment is $15$ and one base is $9$ , then $2(15) = 30$ , and $30 - 9 = 21$ is the other base.

When the lengths are expressions

Sometimes the bases or the midsegment carry variables. Nothing changes: write the same equation, then solve it like any algebra problem. If the bases are $x$ and $x + 6$ and the midsegment is $10$ , then $\dfrac{x + (x+6)}{2} = 10$ , which becomes $2x + 6 = 20$ and gives $x = 7$ .

Two habits keep these clean. Clear the fraction first by multiplying both sides by $2$ , and read the question again at the end, because it may want a length rather than the value of $x$ . Then check your answer by averaging the two bases you found and confirming you land back on the midsegment.

Worked examples

Example 1: find the midsegment

A trapezoid has bases of length $8$ and $14$ . Find the length of its midsegment.

Write the formula

m = \dfrac{b_1 + b_2}{2}

Substitute the two bases

m = \dfrac{8 + 14}{2}

Add, then divide

m = \dfrac{22}{2} = 11

Answer: $m = 11$

Example 2: find a missing base

The midsegment of a trapezoid measures $11$ and one base measures $8$ . Find the other base.

Write the formula with what you know

11 = \dfrac{8 + b_2}{2}

Multiply both sides by

2

22 = 8 + b_2

Subtract

8

from both sides

b_2 = 14

Answer: The other base is $14$ .

Example 3: bases written as expressions

The bases of a trapezoid are $x$ and $x + 6$ , and the midsegment is $10$ . Find the value of $x$ .

Substitute into the formula

10 = \dfrac{x + (x + 6)}{2}

Multiply both sides by

2

20 = 2x + 6

Subtract

6

from both sides

14 = 2x

Divide by

2

x = 7

Answer: $x = 7$ , so the bases are $7$ and $13$ , and their average is $10$ .

Example 4: the midsegment carries the variable

A trapezoid has bases $10$ and $18$ , and its midsegment measures $3x - 1$ . Find $x$ .

Average the bases to get the midsegment's length

\dfrac{10 + 18}{2} = 14

Set the expression equal to that length

3x - 1 = 14

Add

1

to both sides

3x = 15

Divide by

3

x = 5

Answer: $x = 5$

Try one yourself

Common questions

What is the difference between the midsegment of a trapezoid and the midsegment of a triangle?

A triangle's midsegment joins the midpoints of two sides and is half the third side. A trapezoid's midsegment joins the midpoints of the legs and is the average of the two bases. They are really the same idea: a triangle is a trapezoid whose top base has shrunk to $0$ , and $\dfrac{b + 0}{2} = \dfrac{b}{2}$ .

Is the midsegment parallel to the bases?

Yes, always. The midsegment is parallel to both bases, which is part of the theorem and is often the reason a proof uses it. Its length is the average of the two bases.

Can the midsegment be longer than both bases?

No. An average is always between the two numbers being averaged, so the midsegment is longer than the short base and shorter than the long base. If the bases are equal, the midsegment matches them, but then the figure is a parallelogram rather than a trapezoid.

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