Question

point m lies between points l and n on ln. if ln = 12x + 16, what is the length of ln in units? 16 units 40 units 48 units 64 units

Explanation:

Step1: Use the segment addition postulate

Since \(M\) lies between \(L\) and \(N\), \(LM + MN=LN\).
Substitute the given expressions: \((10x + 8)+(5x - 4)=12x + 16\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(10x+5x + 8-4=15x + 4\).
So the equation becomes \(15x + 4=12x + 16\).

Step3: Solve for \(x\)

Subtract \(12x\) from both sides: \(15x-12x + 4=12x-12x + 16\), which gives \(3x+4 = 16\).
Subtract \(4\) from both sides: \(3x+4 - 4=16 - 4\), so \(3x=12\).
Divide both sides by \(3\): \(x=\frac{12}{3}=4\).

Step4: Find the length of \(MN\)

Substitute \(x = 4\) into the expression for \(MN\) (\(MN = 5x-4\)).
\(MN=5\times4-4\).
First, calculate \(5\times4 = 20\), then \(20-4 = 16\). But wait, let's check again.
Wait, no, we made a mistake above. Let's re - check the segment addition.
\(LM+MN = LN\), \(10x + 8+5x - 4=12x + 16\), \(15x+4 = 12x + 16\), \(15x-12x=16 - 4\), \(3x = 12\), \(x = 4\).
Now, \(LN=12x + 16\), substitute \(x = 4\), \(LN=12\times4+16=48 + 16=64\).

Answer:

64 units