Question

in the figure below, triangle ghi is similar to triangle mhl.
what is the distance between m and h?
○ 12
○ 16
○ 24
○ 27

Explanation:

Step1: Recall Similar Triangles Property

For similar triangles \( \triangle GHI \) and \( \triangle MHL \), the corresponding sides are proportional. So, \( \frac{HI}{HL}=\frac{GH}{MH} \).
Given \( HI = 16 \), \( HL = 24 \), \( GH = 18 \), and \( MH = x \).

Step2: Set Up Proportion

Substitute the values into the proportion: \( \frac{16}{24}=\frac{18}{x} \)? Wait, no, wait. Wait, actually, let's check the corresponding sides. Wait, \( \triangle GHI \sim \triangle MHL \), so the sides around the equal angles (vertical angles at \( H \)) should be proportional. So \( \frac{HI}{ML} \)? No, wait, the sides: \( HI \) corresponds to \( ML \)? Wait, no, let's label the triangles. \( \triangle GHI \) has sides \( GH = 18 \), \( HI = 16 \), and \( \triangle MHL \) has sides \( MH = x \), \( HL = 24 \). Wait, actually, the correct proportion is \( \frac{GH}{MH}=\frac{HI}{HL} \)? Wait, no, maybe I mixed up. Wait, \( \triangle GHI \sim \triangle MHL \), so angle at \( H \) is common (vertical angles), so the sides adjacent to angle \( H \) in each triangle should be proportional. So \( \frac{GH}{MH}=\frac{HI}{HL} \). Wait, \( GH = 18 \), \( HI = 16 \), \( HL = 24 \), \( MH = x \). So \( \frac{18}{x}=\frac{16}{24} \)? Wait, no, cross - multiply: \( 16x = 18\times24 \). Wait, no, that can't be. Wait, maybe the correct correspondence is \( \frac{HI}{MH}=\frac{GH}{HL} \). Let's re - establish:

Since \( \triangle GHI \sim \triangle MHL \), the ratio of corresponding sides is equal. So \( \frac{HI}{HL}=\frac{GH}{MH} \)? Wait, \( HI = 16 \), \( HL = 24 \), \( GH = 18 \), \( MH = x \). So \( \frac{16}{24}=\frac{18}{x} \)? No, that gives \( 16x = 18\times24 \), \( x=\frac{18\times24}{16}=\frac{432}{16} = 27 \)? Wait, but 27 is an option. Wait, let's check again.

Wait, maybe the sides are \( HI = 16 \), \( GH = 18 \) in \( \triangle GHI \), and \( HL = 24 \), \( MH = x \) in \( \triangle MHL \). So the proportion is \( \frac{HI}{MH}=\frac{GH}{HL} \). So \( \frac{16}{x}=\frac{18}{24} \). Cross - multiply: \( 18x=16\times24 \), \( x = \frac{16\times24}{18}=\frac{384}{18}=\frac{64}{3}\approx21.33 \). No, that's not right. Wait, maybe I got the correspondence wrong. Wait, the triangles are \( \triangle GHI \) and \( \triangle MHL \), so vertex \( G \) corresponds to \( M \), \( H \) corresponds to \( H \), \( I \) corresponds to \( L \). So \( GH \) corresponds to \( MH \), \( HI \) corresponds to \( HL \), \( GI \) corresponds to \( ML \). So \( \frac{GH}{MH}=\frac{HI}{HL} \). So \( \frac{18}{x}=\frac{16}{24} \). Then \( 16x = 18\times24 \), \( x=\frac{18\times24}{16}=\frac{432}{16}=27 \). Ah, 27 is one of the options.

Answer:

27 (the option with 27)