Question

1. find the perimeter of $\triangle mnp$.

Explanation:

Step1: Use mid - segment theorem

Since $QR$ is a mid - segment of $\triangle MNP$ (by the mid - point markings), $QR=\frac{1}{2}MP$. Given $QR = 25$, then $MP=2\times QR = 50$.

Step2: Use mid - segment theorem for another side

Since $QS$ is a mid - segment of $\triangle MNP$, $QS=\frac{1}{2}NP$. Given $QS = 22$, then $NP = 2\times QS=44$.

Step3: Find the value of $x$

Since $RS$ is a mid - segment of $\triangle MNP$, $RS=\frac{1}{2}MN$. Also, from the figure, we know that $MN=5x - 34$. And $RS=x + 4$. So $2(x + 4)=5x-34$.
Expand the left - hand side: $2x+8 = 5x-34$.
Subtract $2x$ from both sides: $8=3x - 34$.
Add 34 to both sides: $42 = 3x$.
Divide both sides by 3: $x = 14$.

Step4: Calculate the length of $MN$

Substitute $x = 14$ into the expression for $MN$: $MN=5x-34=5\times14 - 34=70 - 34=36$.

Step5: Calculate the perimeter of $\triangle MNP$

The perimeter $P$ of $\triangle MNP$ is $P=MN + NP+MP$.
Substitute $MN = 36$, $NP = 44$, and $MP = 50$: $P=36 + 44+50=130$.

Answer:

130