Question

1. find mn.
2. find jl.

Explanation:

4. Find \( MN \)

Step1: Use the property of perpendicular bisector

Since \( NP\) is the perpendicular bisector of \( MO\), then \( MN = ON\). So, \(4x + 19=6x - 7\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(19 = 2x-7\).
Add \(7\) to both sides: \(2x=19 + 7=26\).
Divide both sides by \(2\): \(x = 13\).

Step3: Find \( MN\)

Substitute \(x = 13\) into \(4x + 19\): \(MN=4\times13+19=52 + 19=71\).

6. Find \( JL\)

Step1: Use the property of mid - segment (since \(KM\) is a mid - segment, \(JM=ML\))

Set \(3x + 4=5x - 16\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(4=2x - 16\).
Add \(16\) to both sides: \(2x=4 + 16=20\).
Divide both sides by \(2\): \(x = 10\).

Step3: Find \( JL\)

\(JL=JM + ML=(3x + 4)+(5x - 16)\). Substitute \(x = 10\): \(JL=(3\times10+4)+(5\times10 - 16)=34 + 34=68\).

Answer:

1. \(MN = 71\)
2. \(JL = 68\)