Question

triangles abc and def are similar. find the length of segment ef.
a 5 cm
b 8.8 cm
c 12 cm
d 14 cm
e 18.75 cm

Explanation:

Step1: Identify corresponding sides

In similar triangles \( \triangle ABC \) and \( \triangle DEF \), the sides \( AB \) and \( DE \) are corresponding, \( BC \) and \( EF \) are corresponding. Given \( AB = 5 \, \text{cm} \), \( DE = 4 \, \text{cm} \), \( BC = 15 \, \text{cm} \).

Step2: Set up proportion for similarity

Since the triangles are similar, the ratios of corresponding sides are equal. So, \( \frac{AB}{DE}=\frac{BC}{EF} \).

Step3: Substitute values and solve for \( EF \)

Substitute \( AB = 5 \), \( DE = 4 \), \( BC = 15 \) into the proportion: \( \frac{5}{4}=\frac{15}{EF} \). Cross - multiply to get \( 5\times EF=4\times15 \). Then \( 5EF = 60 \), and divide both sides by 5: \( EF=\frac{60}{5}=12 \, \text{cm} \)? Wait, no, wait. Wait, maybe I mixed up the corresponding sides. Wait, let's check again. Wait, maybe \( AB \) corresponds to \( DE \), and \( BC \) corresponds to \( EF \)? Wait, no, maybe the sides: \( AB = 5 \), \( DE = 4 \), \( BC = 15 \), \( EF =? \). Wait, maybe the ratio is \( \frac{AB}{DE}=\frac{BC}{EF} \), but let's recalculate. Wait, \( 5/4 = 15/EF \), so \( EF=(15\times4)/5 = 12 \)? But wait, the other side of \( \triangle ABC \) is \( AC = 11 \), but maybe I made a mistake in corresponding sides. Wait, no, maybe \( AB \) is 5, \( DE \) is 4, \( BC \) is 15, so the ratio of similarity is \( 5/4 \). So \( BC \) corresponds to \( EF \), so \( EF = BC\times(4/5)=15\times(4/5)=12 \)? But wait, the options have 12 as option C? Wait, but let's check again. Wait, maybe the corresponding sides are \( AB \) and \( DE \), \( AC \) and \( DF \), \( BC \) and \( EF \). Wait, but let's do the proportion correctly. Wait, \( AB = 5 \), \( DE = 4 \), so the scale factor from \( \triangle ABC \) to \( \triangle DEF \) is \( 4/5 \). So \( EF = BC\times(4/5)=15\times(4/5)=12 \). Wait, but let's check the calculation again. \( 15\times4 = 60 \), \( 60\div5 = 12 \). So \( EF = 12 \, \text{cm} \), which is option C.

Answer:

C. 12 cm