Question

what is the value of w? (triangles are not drawn to scale.) draw diagram of triangles klm and onm with sides: kl=13, lm=10, om=30, mn=w, and arrows indicating parallel sides kl and on, lm and on? show your work here w =

Explanation:

Step1: Identify Similar Triangles

Since \( KL \parallel ON \), \( \triangle KLM \sim \triangle ONM \) by AA similarity (vertical angles at \( M \) and alternate interior angles).

Step2: Set Up Proportion

Corresponding sides of similar triangles are proportional: \( \frac{KM}{OM} = \frac{LM}{NM} \). Substitute \( KM = 13 \), \( OM = 30 \), \( LM = 10 \), \( NM = w \): \( \frac{13}{30} = \frac{10}{w} \)? Wait, no—wait, \( KM \) and \( OM \) are one pair, \( LM \) and \( NM \) are another? Wait, no, correct correspondence: \( KM \) corresponds to \( OM \)? Wait, no, \( K \) corresponds to \( O \), \( L \) to \( N \), \( M \) to \( M \). So \( KM \) (side of \( \triangle KLM \)) corresponds to \( OM \) (side of \( \triangle ONM \))? Wait, no, \( KM = 13 \), \( OM = 30 \), \( LM = 10 \), \( NM = w \). Wait, actually, \( \triangle KLM \sim \triangle ONM \), so \( \frac{KL}{ON} = \frac{KM}{OM} = \frac{LM}{NM} \). Wait, but we have \( KM = 13 \), \( OM = 30 \), \( LM = 10 \), \( NM = w \). So the proportion should be \( \frac{KM}{OM} = \frac{LM}{NM} \)? Wait, no, \( KM \) is from \( K \) to \( M \), \( OM \) from \( O \) to \( M \), \( LM \) from \( L \) to \( M \), \( NM \) from \( N \) to \( M \). So \( \frac{KM}{OM} = \frac{LM}{NM} \) → \( \frac{13}{30} = \frac{10}{w} \)? No, that can't be. Wait, maybe I mixed up the correspondence. Let's re-express: \( \triangle KLM \) and \( \triangle ONM \) are similar, so \( \frac{KM}{OM} = \frac{LM}{NM} \) → \( \frac{13}{30} = \frac{10}{w} \)? Wait, no, that would give \( 13w = 300 \), \( w = 300/13 \), which is not right. Wait, maybe the correct proportion is \( \frac{KM}{NM} = \frac{LM}{OM} \)? Wait, no, let's label the triangles: \( K \)---\( L \), \( O \)---\( N \), with \( M \) as the intersection. So \( KL \parallel ON \), so \( \angle K = \angle O \), \( \angle L = \angle N \), so \( \triangle KLM \sim \triangle ONM \). Therefore, \( \frac{KM}{OM} = \frac{LM}{NM} \). Wait, \( KM = 13 \), \( OM = 30 \), \( LM = 10 \), \( NM = w \). So \( \frac{13}{30} = \frac{10}{w} \)? No, that's inverted. Wait, no—\( KM \) is adjacent to \( \angle M \) in \( \triangle KLM \), \( OM \) is adjacent to \( \angle M \) in \( \triangle ONM \). So \( \frac{KM}{OM} = \frac{LM}{NM} \). So \( 13/30 = 10/w \) → \( w = (30×10)/13 \)? No, that's not right. Wait, maybe I had the sides wrong. Wait, the problem: \( KM = 13 \), \( OM = 30 \), \( LM = 10 \), \( NM = w \). Wait, maybe the correct proportion is \( \frac{KM}{NM} = \frac{LM}{OM} \)? Let's check: \( 13/w = 10/30 \) → \( 10w = 390 \) → \( w = 39 \). Ah, that makes sense! I had the correspondence wrong. So \( \triangle KLM \sim \triangle ONM \), so \( \frac{KM}{NM} = \frac{LM}{OM} \). Because \( KM \) (side of \( \triangle KLM \)) corresponds to \( NM \) (side of \( \triangle ONM \)), and \( LM \) (side of \( \triangle KLM \)) corresponds to \( OM \) (side of \( \triangle ONM \)). Yes, that's the correct correspondence: \( K \) to \( N \), \( L \) to \( O \), \( M \) to \( M \). Wait, no, \( KL \parallel ON \), so \( \angle K = \angle N \), \( \angle L = \angle O \), so \( \triangle KLM \sim \triangle NOM \). Ah! There we go. So \( K \) corresponds to \( N \), \( L \) to \( O \), \( M \) to \( M \). So then \( KM \) (side \( K \) to \( M \)) corresponds to \( NM \) (side \( N \) to \( M \)), \( LM \) (side \( L \) to \( M \)) corresponds to \( OM \) (side \( O \) to \( M \)). So the proportion is \( \frac{KM}{NM} = \frac{LM}{OM} \). So \( \frac{13}{w} = \frac{10}{30} \). Solve for \( w \): cross-multiply: \( 10w = 13×30 \) → \( 10w = 390 \) → \( w = 39 \).

Answer:

\( w = 39 \)