Midsegment Theorem Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

\triangle PQR

M

is the midpoint of

PQ

and

N

is the midpoint of

QR

. If

MN = 3x - 1

and

PR = 4x + 6

, find the value of

x

and the length

MN

1
Step 1: By the Midsegment Theorem, $MN = \frac{1}{2} PR$ . Set up the equation: $3x - 1 = \frac{1}{2}(4x + 6)$ .
2
Step 2: Multiply both sides by 2: $2(3x - 1) = 4x + 6$ , giving $6x - 2 = 4x + 6$ .
3
Step 3: Solve: $2x = 8$ , so $x = 4$ .
4
Step 4: $MN = 3(4) - 1 = 11$ . Check: $PR = 4(4) + 6 = 22$ , and $\frac{22}{2} = 11$ . ✓

Answer

x = 4

;

MN = 11

The Midsegment Theorem gives a direct equation relating the midsegment to the parallel side: midsegment

= \frac{1}{2} \times

parallel side. Substituting the algebraic expressions and solving for

x

is the standard algebraic application of this theorem.

About Midsegment Theorem

A segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.

In [formula], [formula] is the midpoint of [formula] and [formula] is the midpoint of [formula]. If

The midsegment of a triangle has length 14. What is the length of the side parallel to the midsegmen

In [formula], the three midsegments are drawn, dividing the triangle into four smaller triangles. If