Question

5 fill in the blank 1 point
find the perimeter of \\( \triangle m n p \\).

\\( x = \\) type your answer...
perimeter of \\( \triangle m n p = \\) type your answer...

Explanation:

Step1: Use the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length.
For the mid - segment \(QR\) (where \(QR = 25\)), if \(QR\) is a mid - segment of \(\triangle MNP\), then \(MP=2QR\).
For the mid - segment \(QS\) (where \(QS = 22\)), if \(QS\) is a mid - segment of \(\triangle MNP\), then \(NP = 2QS\).
For the mid - segment \(RS\) (where \(RS=x + 4\)), if \(RS\) is a mid - segment of \(\triangle MNP\), then \(MN=2RS\). Also, \(MN = 5x-34\). So, \(5x-34=2(x + 4)\)

Step2: Solve the equation \(5x-34=2(x + 4)\)

Expand the right - hand side: \(5x-34=2x+8\)
Subtract \(2x\) from both sides: \(5x-2x-34=2x-2x + 8\), \(3x-34=8\)
Add \(34\) to both sides: \(3x-34 + 34=8+34\), \(3x=42\)
Divide both sides by \(3\): \(x=\frac{42}{3}=14\)

Step3: Find the lengths of the sides of \(\triangle MNP\)

\(MN=5x-34\), substitute \(x = 14\): \(MN=5\times14-34=70 - 34=36\)
\(NP=2\times22 = 44\)
\(MP=2\times25=50\)

Step4: Calculate the perimeter of \(\triangle MNP\)

The perimeter \(P\) of a triangle is \(P=MN + NP+MP\)
\(P=36 + 44+50=130\)

Answer:

\(x = 14\)
Perimeter of \(\triangle MNP=130\)