Question

33,980 Learners found this answer helpful

1. δabc is similar to δdef. the lengths of the sides of δabc are 5, 8, and 11. what is the length of the shortest side of δdef if its perimeter is 60?

Explanation:

Step1: Find perimeter of ΔABC

The sides of ΔABC are 5, 8, 11. So perimeter \( P_{ABC} = 5 + 8 + 11 = 24 \).

Step2: Determine scale factor

Let the scale factor be \( k \). The perimeter of ΔDEF is 60. Since similar triangles have perimeters in the ratio of their scale factor, \( k=\frac{P_{DEF}}{P_{ABC}}=\frac{60}{24}=\frac{5}{2} \). Wait, no—wait, actually, we need to find the shortest side of ΔDEF. The shortest side of ΔABC is 5. For similar triangles, corresponding sides are in proportion (scale factor). Wait, maybe I mixed up. Wait, the ratio of perimeters is equal to the ratio of corresponding sides. So let the shortest side of ΔDEF be \( x \). Then \( \frac{x}{5}=\frac{60}{24} \). Let's recalculate perimeter of ABC: 5+8+11=24. Perimeter of DEF is 60. So the scale factor from ABC to DEF is \( \frac{60}{24}=\frac{5}{2} \)? Wait, no—if ABC ~ DEF, then \( \frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}}=\frac{\text{Side of ABC}}{\text{Side of DEF}} \). Wait, no, similar triangles: the ratio of perimeters is equal to the ratio of corresponding sides. So if ABC ~ DEF, then \( \frac{P_{ABC}}{P_{DEF}}=\frac{\text{Shortest side of ABC}}{\text{Shortest side of DEF}} \)? Wait, no, actually, if ABC ~ DEF, then \( \frac{\text{Side of ABC}}{\text{Side of DEF}}=\frac{P_{ABC}}{P_{DEF}} \). Wait, let's define: let \( a = 5 \) (shortest of ABC), \( A = x \) (shortest of DEF), \( P_{ABC}=24 \), \( P_{DEF}=60 \). Then \( \frac{a}{A}=\frac{P_{ABC}}{P_{DEF}} \)? No, that would be inverse. Wait, no: similar triangles, the ratio of sides is equal to the ratio of perimeters. So \( \frac{\text{Side of ABC}}{\text{Side of DEF}}=\frac{P_{ABC}}{P_{DEF}} \) is wrong. Wait, correct formula: if ΔABC ~ ΔDEF, then \( \frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}=\frac{P_{ABC}}{P_{DEF}} \). So the ratio of corresponding sides is equal to the ratio of their perimeters. So the shortest side of ABC is 5, so let the shortest side of DEF be \( x \). Then \( \frac{5}{x}=\frac{24}{60} \)? Wait, no, that's the mistake. Wait, \( \frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}}=\frac{\text{Side of ABC}}{\text{Side of DEF}} \). So \( \frac{24}{60}=\frac{5}{x} \)? Solving for \( x \): \( 24x = 5×60 \), \( 24x=300 \), \( x=\frac{300}{24}=\frac{25}{2}=12.5 \)? Wait, no, that can't be. Wait, no—wait, maybe I had the ratio reversed. If DEF has a larger perimeter, then the scale factor is \( \frac{60}{24}=\frac{5}{2} \), so each side of DEF is \( \frac{5}{2} \) times the corresponding side of ABC. So shortest side of ABC is 5, so shortest side of DEF is \( 5×\frac{5}{2}=12.5 \)? Wait, but 5×(60/24)=5×(5/2)=12.5. Let's check: perimeter of ABC is 24, perimeter of DEF would be 5×(5/2) + 8×(5/2) + 11×(5/2) = (5+8+11)×(5/2)=24×(5/2)=60. Yes, that works. So the shortest side of DEF is 5×(60/24)=12.5, which is \( \frac{25}{2} \) or 12.5.

Wait, let's redo the steps correctly:

1. Calculate perimeter of ΔABC: \( 5 + 8 + 11 = 24 \).
2. Let the shortest side of ΔDEF be \( x \). Since ΔABC ~ ΔDEF, the ratio of their corresponding sides is equal to the ratio of their perimeters. So \( \frac{\text{Shortest side of ABC}}{\text{Shortest side of DEF}}=\frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}} \) is incorrect. Wait, no: the ratio of sides is equal to the ratio of perimeters. So \( \frac{\text{Side of ABC}}{\text{Side of DEF}}=\frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}} \) is wrong. Wait, actually, \( \frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}}=\frac{\text{Length of side in ABC}}{\text{Length of corresponding side in DEF}} \). So if ABC ~ DEF, then \( \frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}=\frac{P_{ABC}}{P_{DEF}} \). So let AB = 5 (shortest in ABC), DE = x (shortest in DEF). Then \( \frac{AB}{DE}=\frac{P_{ABC}}{P_{DEF}} \) → \( \frac{5}{x}=\frac{24}{60} \) → \( x=\frac{5×60}{24}=\frac{300}{24}=12.5 \). Yes, that's correct. So the shortest side of DEF is 12.5 (or \( \frac{25}{2} \)).

Answer:

The length of the shortest side of \( \triangle DEF \) is \( \boldsymbol{12.5} \) (or \( \boldsymbol{\frac{25}{2}} \)).