Question

what is the length of segment lm?
□ units
(diagram with triangle k n m, point l on k m, perpendicular at l, k l = 9x + 5, k n = 14x - 3, n m = 25, and line a through n and l)

Explanation:

Step1: Identify congruent segments

Since \( LN \perp KM \) and \( K N = N M = 25 \) (marked congruent), triangle \( K N M \) is isosceles with \( LN \) as the perpendicular bisector. So \( KL = LM \), and \( K N = 14x - 3 = 25 \).

Step2: Solve for \( x \)

Set \( 14x - 3 = 25 \). Add 3 to both sides: \( 14x = 28 \). Divide by 14: \( x = 2 \).

Step3: Find \( LM \)

\( LM = 9x + 5 \). Substitute \( x = 2 \): \( 9(2) + 5 = 18 + 5 = 23 \)? Wait, no—wait, \( KL = LM \), but \( K N = N M = 25 \), and \( KL = 9x + 5 \), \( K N = 14x - 3 = 25 \). Wait, maybe I mixed up. Wait, \( K N = 14x - 3 = 25 \), so \( x = 2 \). Then \( KL = 9x + 5 = 9(2) + 5 = 23 \), but \( LM = KL \)? Wait, no, the diagram: \( LN \) is perpendicular to \( KM \), so \( L \) is the midpoint? Wait, no, \( K N = N M = 25 \), so triangle \( K N M \) has \( K N = N M \), so it's isosceles with \( K N = N M \), and \( LN \perp KM \), so \( L \) is the foot, so \( KL = LM \). Wait, but \( K N = 14x - 3 = 25 \), so \( x = 2 \), then \( KL = 9x + 5 = 23 \), so \( LM = 23 \)? Wait, but \( N M = 25 \), that seems conflicting. Wait, maybe \( K N = N M = 25 \), so \( 14x - 3 = 25 \), so \( x = 2 \). Then \( KL = 9x + 5 = 23 \), and \( LM = KL = 23 \)? Wait, but the problem is to find \( LM \). Wait, maybe I made a mistake. Wait, let's re-express:

Given \( K N = 14x - 3 \), \( N M = 25 \), and since \( LN \) is perpendicular to \( KM \) and \( K N = N M \) (marked congruent), so triangle \( K N M \) is isosceles with \( K N = N M \), so \( 14x - 3 = 25 \). Solve for \( x \): \( 14x = 28 \), \( x = 2 \). Then \( LM = 9x + 5 = 9(2) + 5 = 23 \). Wait, but \( KL = LM \), so \( LM = 23 \). Wait, but let's check again.

Wait, maybe the segments: \( K L = 9x + 5 \), \( L M =? \), and \( K N = 14x - 3 \), \( N M = 25 \). Since \( LN \perp KM \), and \( K N = N M \), so \( L \) is the midpoint? No, \( K N = N M \) means \( N \) is equidistant from \( K \) and \( M \), so \( LN \) is the perpendicular bisector, so \( L \) is the midpoint of \( KM \), so \( KL = LM \). Therefore, \( KL = 9x + 5 \), \( LM = 9x + 5 \), and \( K N = 14x - 3 = 25 \). So \( x = 2 \), then \( LM = 9(2) + 5 = 23 \). Wait, but \( N M = 25 \), that's a side, not \( LM \). So the answer is 23? Wait, no, maybe I messed up the labels. Wait, the diagram: \( K \) to \( N \) is \( 14x - 3 \), \( N \) to \( M \) is 25, \( K \) to \( L \) is \( 9x + 5 \), \( L \) to \( M \) is what we need. Since \( LN \perp KM \), and \( K N = N M \), so triangle \( K N M \) is isosceles with \( K N = N M \), so \( 14x - 3 = 25 \), so \( x = 2 \). Then \( K L = 9x + 5 = 23 \), and since \( L \) is the foot of the perpendicular from \( N \) to \( KM \) in isosceles triangle \( K N M \), \( L \) bisects \( KM \), so \( K L = L M \), so \( L M = 23 \). Yes, that makes sense.

Answer:

23