Question

find the perimeter of $\triangle mnp$
answer attempt 3 out of 99
$mp = \square$
$np = \square$
$x = \square$
$mn = \square$
$perimeter = \square$

Explanation:

Step1: Identify Midsegments and Solve for \( x \)

In triangle \( MNP \), \( Q \), \( R \), \( S \) are midpoints (since segments are marked congruent). So \( QS \) and \( RS \) are midsegments. For midsegment \( RS \), \( RS = \frac{1}{2}MN \). Given \( RS = x + 4 \) and \( MN = 5x - 34 \), we have \( x + 4=\frac{1}{2}(5x - 34) \). Multiply both sides by 2: \( 2x + 8 = 5x - 34 \). Subtract \( 2x \): \( 8 = 3x - 34 \). Add 34: \( 42 = 3x \), so \( x = 14 \).

Step2: Find \( MN \)

Substitute \( x = 14 \) into \( MN = 5x - 34 \): \( MN = 5(14)-34 = 70 - 34 = 36 \).

Step3: Find \( MP \)

\( S \) is midpoint of \( MP \), and \( MS = 22 \), so \( MP = 2 \times 22 = 44 \).

Step4: Find \( NP \)

\( R \) is midpoint of \( NP \), and \( QR = 25 \) (midsegment, so \( QR=\frac{1}{2}MP \)? Wait, no, \( QS \) is midsegment? Wait, \( Q \) is midpoint of \( MN \), \( S \) midpoint of \( MP \), so \( QS \parallel NP \) and \( QS=\frac{1}{2}NP \). Wait, \( QS = 25 \)? Wait, no, the segment \( QR = 25 \)? Wait, maybe I mixed up. Wait, \( Q \) midpoint of \( MN \), \( R \) midpoint of \( NP \), \( S \) midpoint of \( MP \). So \( QR \parallel MP \) and \( QR=\frac{1}{2}MP \). \( MP = 44 \), so \( QR=\frac{1}{2}\times44 = 22 \)? No, the diagram has \( QS = 25 \)? Wait, maybe \( QS \) is midsegment to \( NP \), so \( QS=\frac{1}{2}NP \). So \( NP = 2 \times 25 = 50 \).

Step5: Calculate Perimeter

Perimeter of \( \triangle MNP \) is \( MN + MP + NP = 36 + 44 + 50 = 130 \).

Answer:

\( MP = 44 \)
\( NP = 50 \)
\( x = 14 \)
\( MN = 36 \)
\( \text{Perimeter} = 130 \)