Question

1. find the perimeter of \\( \triangle m n p \\).

Explanation:

Step1: Use the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length.
For example, if \(QR\) is a mid - segment, then \(QR=\frac{1}{2}MP\), \(QS=\frac{1}{2}NP\), \(SR=\frac{1}{2}MN\).
Since \(QR = 25\), then \(MP=2\times25 = 50\).
Since \(QS = 22\), then \(NP=2\times22=44\).
Since \(SR=x + 4\), and \(MN = 5x-34\), and \(SR=\frac{1}{2}MN\), we have the equation \(x + 4=\frac{1}{2}(5x-34)\).
Multiply both sides by 2: \(2(x + 4)=5x-34\).
Expand: \(2x+8 = 5x-34\).
Subtract \(2x\) from both sides: \(8=3x-34\).
Add 34 to both sides: \(42 = 3x\).
Divide by 3: \(x = 14\).
Then \(MN=5x-34=5\times14 - 34=70 - 34=36\).

Step2: Calculate the perimeter of \(\triangle MNP\)

The perimeter \(P\) of \(\triangle MNP\) is \(P=MN + NP+MP\).
Substitute \(MN = 36\), \(NP = 44\), \(MP = 50\) into the formula.
\(P=36 + 44+50\).
\(P=130\).

Answer:

\(130\)