Question

1. find the perimeter of \\( \triangle m n p \\).
2. find the measure of each missing angle

Explanation:

Step1: Use the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length.
For triangle \(MNP\), \(QR\) is a mid - segment. If \(QR = 25\), then \(MP=2\times QR\). So \(MP = 50\).
\(QS\) is a mid - segment. If \(QS = 22\), then \(NP=2\times QS\). So \(NP = 44\).
\(RS\) is a mid - segment. If \(RS=x + 4\), then \(MN=2\times RS\). Also, since \(Q\) is the mid - point of \(MN\), \(MQ=QN\). Given \(MQ = 5x-34\) and \(QN = RS\times2-(5x - 34)\) (because \(MN = 2RS\)).
Since \(MQ=\frac{1}{2}MN\) and \(MN = 2(x + 4)\), we have \(5x-34=x + 4\).
Solve the equation \(5x-34=x + 4\):
Subtract \(x\) from both sides: \(5x-x-34=x - x+ 4\), \(4x-34 = 4\).
Add \(34\) to both sides: \(4x-34 + 34=4 + 34\), \(4x=38\), \(x=\frac{38}{4}=\frac{19}{2}\). Then \(RS=\frac{19}{2}+ 4=\frac{19 + 8}{2}=\frac{27}{2}\), and \(MN=2\times\frac{27}{2}=27\).

Step2: Calculate the perimeter

The perimeter \(P\) of \(\triangle MNP\) is \(P=MN + NP+MP\).
Substitute \(MN = 27\), \(NP = 44\), and \(MP = 50\) into the formula: \(P=27 + 44+50\).
\(P=121\).

Answer:

The perimeter of \(\triangle MNP\) is \(121\).