Question

mn is the midsegment of qrs.
solve for x and mn.
$qs = 3x + 5$
$mn = 2x - 1$
$x = \underline{\quad\quad}$
$mn = \underline{\quad\quad}$

Explanation:

Step1: Apply Midsegment Theorem

The midsegment of a triangle is half the length of the third side, so $MN = \frac{1}{2}QS$. Substitute the given expressions:
$2x - 1 = \frac{1}{2}(3x + 5)$

Step2: Eliminate the fraction

Multiply both sides by 2 to clear the denominator:
$2(2x - 1) = 3x + 5$
$4x - 2 = 3x + 5$

Step3: Solve for x

Subtract $3x$ from both sides, then add 2 to both sides:
$4x - 3x = 5 + 2$
$x = 7$

Step4: Calculate MN

Substitute $x=7$ into the expression for $MN$:
$MN = 2(7) - 1$
$MN = 14 - 1 = 13$

Answer:

$x = 7$
$MN = 13$