Question

ln is an angle bisector of <klm. what is the length on mn?

Explanation:

Step1: Use the Angle Bisector Theorem for Congruent Segments

Since \(LN\) is an angle bisector and \(\angle LKN=\angle LMN = 90^{\circ}\), by the Angle - Bisector Theorem (in the form of congruent right - triangles \( \triangle LKN\) and \( \triangle LMN\) because \(LN = LN\) (common side), \(\angle KLN=\angle MLN\), \(\angle LKN=\angle LMN\)), we have \(KN = MN\). So \(7x−4=5x + 12\).

Step2: Solve the Equation for \(x\)

Subtract \(5x\) from both sides: \(7x-5x−4=5x - 5x+12\), which gives \(2x−4 = 12\).
Add \(4\) to both sides: \(2x-4 + 4=12 + 4\), so \(2x=16\).
Divide both sides by \(2\): \(x=\frac{16}{2}=8\).

Step3: Find the Length of \(MN\)

Substitute \(x = 8\) into the expression for \(MN\) (\(MN=5x + 12\)).
\(MN=5\times8+12\).
First, calculate \(5\times8 = 40\), then \(40+12=52\).

Answer:

\(52\)