Question

if jl = 30, jk = 18, and lm = 6, then the value of ln is.
3
6
9
15

Explanation:

Step1: Identify Similar Triangles

From the diagram, \( KM \parallel JN \) (implied by the congruent marks on the sides), so \( \triangle JLN \sim \triangle KLM \) by the Basic Proportionality Theorem (Thales' theorem). Thus, the ratios of corresponding sides are equal: \( \frac{JK}{KL}=\frac{LM}{MN} \)? Wait, no, correct proportion: \( \frac{JL}{KL}=\frac{LN}{LM} \)? Wait, let's re - establish the sides. Given \( JL = 30 \), \( JK = 18 \), so \( KL=JL - JK=30 - 18 = 12 \). Let \( LN=x \), then \( MN = LN - LM=x - 6 \)? Wait, no, better: Since \( KM\parallel JN \), \( \frac{JK}{KL}=\frac{MN}{LM} \)? Wait, maybe the correct proportion is \( \frac{JK}{JL}=\frac{LM}{LN} \)? Wait, no, let's use the similarity of triangles. If \( \triangle JLN\sim\triangle KLM \), then \( \frac{JK}{JL}=\frac{LM}{LN} \)? Wait, \( JK = 18 \), \( JL = 30 \), \( LM = 6 \), let \( LN=x \). Then \( \frac{JK}{JL}=\frac{LM}{LN}\)? Wait, no, \( \frac{KL}{JL}=\frac{LM}{LN} \), and \( KL=JL - JK = 30 - 18=12 \). So \( \frac{12}{30}=\frac{6}{x} \)? No, that gives \( 12x = 180\), \( x = 15 \). Wait, let's check again. The correct proportion from similar triangles (by the theorem of parallel lines cutting sides proportionally) is \( \frac{JK}{KL}=\frac{MN}{LM} \)? No, the correct setup is: In \( \triangle JLN \), \( KM\parallel JN \), so \( \frac{JK}{KL}=\frac{MN}{LM} \)? Wait, no, the segments on \( JL \) are \( JK = 18 \) and \( KL=30 - 18 = 12 \), and on \( LN \) are \( LM = 6 \) and \( MN \). So by the Basic Proportionality Theorem, \( \frac{JK}{KL}=\frac{MN}{LM} \)? No, the theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So the line \( KM \) is parallel to \( JN \), intersecting \( JL \) at \( K \) and \( LN \) at \( M \). So \( \frac{JK}{KL}=\frac{MN}{LM} \)? Wait, no, \( \frac{JK}{JL}=\frac{LM}{LN} \). Let's substitute values: \( JK = 18 \), \( JL = 30 \), \( LM = 6 \), \( LN=x \). Then \( \frac{18}{30}=\frac{6}{x} \)? Cross - multiply: \( 18x=30\times6 = 180 \), so \( x = 10 \)? Wait, that's wrong. Wait, I think I mixed up the sides. Let's define the sides properly. Let \( LN=x \), so \( LN=LM + MN=6 + MN \), so \( MN=x - 6 \). The sides on \( JL \): \( JK = 18 \), \( KL = 30 - 18 = 12 \). By the proportionality theorem, \( \frac{JK}{KL}=\frac{MN}{LM} \), so \( \frac{18}{12}=\frac{x - 6}{6} \). Cross - multiply: \( 18\times6=12\times(x - 6) \), \( 108 = 12x-72 \), \( 12x=180 \), \( x = 15 \). Yes, that works. So \( LN = 15 \).

Answer:

15