Question

find w.

diagram of two right triangles, triangle jim with right angle at j, ji = 32 ft, jm = 60 ft; triangle klm with right angle at l, kl = 16 ft, ml = w; angles at m are equal (vertical angles) and angles at j and l are right angles

write your answer as a whole number or a decimal. do not round.
w = \boxed{} feet

submit

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle IJM \) and \( \triangle KLM \) are similar (AA similarity: right angles and vertical angles at \( M \)).

Step2: Set Up Proportion

Since similar triangles have proportional sides, \( \frac{IJ}{KL} = \frac{JM}{LM} \). Substitute \( IJ = 32 \), \( KL = 16 \), \( JM = 60 \), \( LM = w \). So \( \frac{32}{16} = \frac{60}{w} \).

Step3: Solve for \( w \)

Simplify \( \frac{32}{16} = 2 \), so \( 2 = \frac{60}{w} \). Cross - multiply: \( 2w = 60 \), then \( w=\frac{60}{2}=30 \). Wait, no, wait. Wait, actually, the triangles are \( \triangle IJM \) and \( \triangle KLM \), but also, \( \triangle IJM \sim \triangle KLM \), but \( JM \) is 60, and \( LM \) is \( w \), but also, the ratio of \( IJ \) to \( KL \) is \( 32:16 = 2:1 \). So the ratio of corresponding sides should be equal. Wait, maybe I mixed up the triangles. Let's re - examine. The right triangles: \( \angle J=\angle L = 90^{\circ} \), \( \angle IMJ=\angle KML \) (vertical angles), so \( \triangle IJM \sim \triangle KLM \). So \( \frac{IJ}{KL}=\frac{JM}{LM} \). \( IJ = 32 \), \( KL = 16 \), \( JM = 60 \), \( LM = w \). So \( \frac{32}{16}=\frac{60}{w} \)? Wait, no, that would give \( w = 30 \), but let's check again. Wait, maybe the triangles are \( \triangle IJM \) and \( \triangle KLM \), but actually, \( IJ = 32 \), \( KL = 16 \), so the ratio of similarity is \( 32/16 = 2 \). So the sides of \( \triangle IJM \) are twice the sides of \( \triangle KLM \). So \( JM \) should be twice \( LM \). So \( 60 = 2w \), so \( w = 30 \)? Wait, no, wait, maybe I got the correspondence wrong. Let's see: \( \triangle IJM \) has legs \( IJ = 32 \) and \( JM = 60 \), \( \triangle KLM \) has legs \( KL = 16 \) and \( LM = w \). Since they are similar, \( \frac{IJ}{KL}=\frac{JM}{LM} \), so \( \frac{32}{16}=\frac{60}{w} \), \( 2=\frac{60}{w} \), \( w = 30 \). Wait, but let's check with another approach. The ratio of \( IJ \) to \( KL \) is \( 32:16 = 2:1 \), so the ratio of \( JM \) to \( LM \) should also be \( 2:1 \). So \( JM = 2\times LM \), \( 60 = 2w \), so \( w = 30 \).

Answer:

\( w = 30 \)