Question

triangle similarity theorems
what is the perimeter of △aeb?
22.9 ft
18.7 ft
18.5 ft
16.4 ft

Explanation:

Step1: Identify similar triangles

Since \( DF \parallel AB \) (marked with equal arrows), \( \triangle DFE \sim \triangle ABE \) by AA similarity.

Step2: Find the scale factor

The ratio of corresponding sides \( \frac{DA}{AF} \)? Wait, no, \( DA = 2 \) ft, \( AF \)? Wait, \( DF \) and \( AB \) are parallel, so \( \frac{DF}{AB}=\frac{EF}{EB} \)? Wait, \( DF \) length: from \( D \) to \( F \), but \( AB = 4.2 \) ft, \( DF \) is equal to \( AB \)? Wait, no, the segments: \( DA = 2 \) ft, \( FB = 2.4 \) ft? Wait, maybe the ratio of \( DA \) to \( AF \)? Wait, no, let's look at the sides. The length from \( D \) to \( A \) is 2 ft, and from \( F \) to \( B \) is 2.4 ft? Wait, maybe the ratio of \( DA \) to \( FB \)? Wait, no, the triangles \( \triangle DFE \) and \( \triangle ABE \): \( DA = 2 \) ft, \( AF \) is not given, but \( AB = 4.2 \) ft, \( EF = EB + 7.8 \)? Wait, no, \( EB = 7.8 \) ft? Wait, the length of \( EB \) is 7.8 ft? Wait, no, the diagram: \( E \) to \( B \) is 7.8 ft? Wait, maybe the ratio of \( DA \) to \( DF \)? Wait, no, let's re - examine.

Wait, the sides \( DA = 2 \) ft, \( FB = 2.4 \) ft? Wait, no, the horizontal segment: \( DA = 2 \) ft, and the segment from \( F \) to \( B \) is 2.4 ft? Wait, maybe the ratio of \( DA \) to \( FB \) is \( \frac{2}{2.4}=\frac{5}{6} \)? Wait, no, maybe the ratio of similarity is \( \frac{DA}{DA + AF} \)? No, this is confusing. Wait, another approach: since \( \triangle DFE \sim \triangle ABE \), the ratio of \( DA \) to \( AF \) is not, but \( DA = 2 \) ft, \( AB = 4.2 \) ft, \( EF = EB + 7.8 \)? Wait, no, the length of \( EB \) is 7.8 ft? Wait, the problem is to find the perimeter of \( \triangle AEB \). Let's find the lengths of \( AE \), \( AB \), and \( EB \). Wait, \( EB = 7.8 \) ft? No, \( EB \) is part of \( EF \). Wait, maybe the ratio of \( DA \) to \( FB \) is \( \frac{2}{2.4}=\frac{5}{6} \), so the ratio of similarity between \( \triangle DFE \) and \( \triangle ABE \) is \( \frac{5}{6} \). Then, if \( EF = EB + 7.8 \), and \( \frac{EF}{EB}=\frac{6}{5} \) (since \( \triangle DFE \) is larger), so \( \frac{EB + 7.8}{EB}=\frac{6}{5} \), cross - multiply: \( 5(EB + 7.8)=6EB \), \( 5EB+39 = 6EB \), \( EB = 39 \)? No, that can't be. Wait, maybe I misread the diagram.

Wait, the length of \( EB \) is 7.8 ft? No, the diagram shows \( EB = 7.8 \) ft? Wait, no, the segment from \( E \) to \( B \) is 7.8 ft? Wait, maybe the sides: \( AB = 4.2 \) ft, \( EB = 7.8 \) ft? No, that doesn't make sense. Wait, let's start over.

Since \( DF \parallel AB \), \( \triangle DFE \sim \triangle ABE \). The ratio of \( DA \) to \( AF \) is \( \frac{2}{2.4}=\frac{5}{6} \)? Wait, \( DA = 2 \) ft, \( FB = 2.4 \) ft? Wait, \( DA \) and \( FB \) are corresponding segments? Maybe \( DA = 2 \) ft, \( AB = 4.2 \) ft, and the ratio of \( DA \) to \( AB \) is not. Wait, the perimeter of \( \triangle DFE \) would be \( DA + DF + EF \), but we need \( \triangle AEB \). Wait, maybe the ratio of similarity is \( \frac{DA}{DA + AF} \)? No, this is wrong.

Wait, another way: the length of \( AE \): let's assume that the ratio of \( DA \) to \( FB \) is \( \frac{2}{2.4}=\frac{5}{6} \), so the ratio of \( \triangle AEB \) to \( \triangle DFE \) is \( \frac{5}{6} \). The length of \( EF = EB + 7.8 \), so if \( \frac{EB}{EF}=\frac{5}{6} \), then \( \frac{EB}{EB + 7.8}=\frac{5}{6} \), \( 6EB = 5EB + 39 \), \( EB = 39 \)? No, that's too big. I must have misread the diagram.

Wait, maybe the length of \( EB \) is 7.8 ft, and we need to find \( AE \) and \( AB \). Wait, \( AB = 4.2 \) ft, \( EB = 7.8 \) ft, and we need to find \( AE \). Using the similar triangles: \( \frac{DA}{AB}=\frac{AE}{EF} \)? No, \( DA = 2 \) ft, \( AB = 4.2 \) ft, \( FB = 2.4 \) ft. The ratio of \( DA \) to \( FB \) is \( \frac{2}{2.4}=\frac{5}{6} \), so the ratio of \( \triangle AEB \) to \( \triangle DFE \) is \( \frac{5}{6} \). The perimeter of \( \triangle DFE \) would be \( DA + DF + EF \), but \( DF = AB = 4.2 \) ft, \( DA = 2 \) ft, \( FB = 2.4 \) ft, \( EF = EB + 7.8 \). Wait, this is too confusing. Maybe the correct approach is:

Since \( \triangle DFE \sim \triangle ABE \), the ratio of corresponding sides is \( \frac{DA}{DA + AF} \)? No, \( DA = 2 \) ft, \( AF \) is not given. Wait, maybe the length of \( AE \): let's use the ratio of \( 2 \) to \( 2 + 2.4=4.4 \)? No, that's not. Wait, the answer choices are 22.9, 18.7, 18.5, 16.4. Let's calculate the perimeter of \( \triangle AEB \) as \( AB + EB + AE \). \( AB = 4.2 \) ft, \( EB = 7.8 \) ft, so we need \( AE \). Using similar triangles: \( \frac{DA}{FB}=\frac{AE}{EB} \)? \( DA = 2 \), \( FB = 2.4 \), \( EB = 7.8 \), so \( AE=\frac{2\times7.8}{2.4}=\frac{15.6}{2.4}=6.5 \) ft. Then the perimeter is \( 4.2+7.8 + 6.5=18.5 \) ft.

Step3: Calculate the perimeter

Perimeter of \( \triangle AEB=AB + EB+AE \). We found \( AE = 6.5 \) ft, \( AB = 4.2 \) ft, \( EB = 7.8 \) ft. So \( 4.2+7.8 + 6.5 = 18.5 \) ft.

Answer:

18.5 ft