Question

1. △mnp is isosceles with $overline{mn}congoverline{np}$. mn is two less than five times x. np is seven more than two times x. pm is two more than three times x. find x and the length of each side.

x =
mn =
np =
mp =

Explanation:

Step1: Set up the equation

Since $\overline{MN}\cong\overline{NP}$, we set $5x - 2=2x + 7$.

Step2: Solve for x

Subtract $2x$ from both sides: $5x-2x - 2=2x-2x + 7$, which simplifies to $3x-2 = 7$. Then add 2 to both sides: $3x-2 + 2=7 + 2$, getting $3x=9$. Divide both sides by 3: $x=\frac{9}{3}=3$.

Step3: Find the length of MN

Substitute $x = 3$ into the expression for $MN$. $MN=5x - 2=5\times3-2=15 - 2=13$.

Step4: Find the length of NP

Since $MN = NP$, $NP=13$.

Step5: Find the length of MP

Substitute $x = 3$ into the expression for $MP$. $MP=3x + 2=3\times3+2=9 + 2=11$.

Answer:

$x = 3$
$MN = 13$
$NP = 13$
$MP = 11$