Answer :
Given the problem, we need to determine which of the provided sets of side lengths form triangles that are similar to a triangle with side lengths 7, 24, and 25. Two triangles are considered similar if their corresponding sides are in the same ratio.
The original Pythagorean triple: 7, 24, 25
Here is the step-by-step process to determine the similarity for each set:
1. Set: 14, 48, 50
Let's see if the sides are proportional to the original triple:
[tex]\[ \frac{14}{7} = 2, \quad \frac{48}{24} = 2, \quad \frac{50}{25} = 2 \][/tex]
All ratios are equal (2), so this triangle is similar to the 7, 24, 25 triangle.
- Answer: Yes
2. Set: 9, 12, 15
Let's check the ratios:
[tex]\[ \frac{9}{7} \approx 1.29, \quad \frac{12}{24} = 0.5, \quad \frac{15}{25} = 0.6 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
3. Set: [tex]\(2, \sqrt{20}, 2 \sqrt{6}\)[/tex]
Let's check the ratios:
[tex]\[ \frac{2}{7} \approx 0.29, \quad \frac{\sqrt{20}}{24} \approx 0.21, \quad \frac{2\sqrt{6}}{25} \approx 0.31 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
4. Set: 8, 15, 17
Let's check the ratios:
[tex]\[ \frac{8}{7} \approx 1.14, \quad \frac{15}{24} = 0.625, \quad \frac{17}{25} = 0.68 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
5. Set: [tex]\(\sqrt{7}, \sqrt{24}, \sqrt{25}\)[/tex]
Let's check the ratios:
[tex]\[ \frac{\sqrt{7}}{7}, \quad \frac{\sqrt{24}}{24}, \quad \frac{\sqrt{25}}{25} \][/tex]
[tex]\(\frac{\sqrt{7}}{7}\)[/tex] and [tex]\(\frac{\sqrt{24}}{24}: \sqrt{24} \approx 4.9, \frac{4.9}{24} \neq \frac{7}{7}\)[/tex]
[tex]\(\frac{\sqrt{25}}{25}: \frac{5}{25} \neq 1\)[/tex]
These ratios are not equal, so this triangle is not similar.
- Answer: No
6. Set: 35, 120, 125
Let's check the ratios:
[tex]\[ \frac{35}{7} = 5, \quad \frac{120}{24} = 5, \quad \frac{125}{25} = 5 \][/tex]
All ratios are equal (5), so this triangle is similar to the 7, 24, 25 triangle.
- Answer: Yes
7. Set: 21, 72, 78
Let's check the ratios:
[tex]\[ \frac{21}{7} = 3, \quad \frac{72}{24} = 3, \quad \frac{78}{25} \neq 3 \][/tex]
The ratios are not all equal, so this triangle is not similar (However, consideration may lead to a Yes answer from examining ratios - given constraints).
- Answer: No
After completing the analysis for each set, we conclude the sets of side lengths that form triangles similar to the triangle with side lengths 7, 24, and 25 are:
- [tex]\(14, 48, 50\)[/tex]
- \(35, 120, 125)
So, the correct answers are:
[tex]\[ \boxed{14, 48, 50 \text{ and } 35, 120, 125} \][/tex]
The original Pythagorean triple: 7, 24, 25
Here is the step-by-step process to determine the similarity for each set:
1. Set: 14, 48, 50
Let's see if the sides are proportional to the original triple:
[tex]\[ \frac{14}{7} = 2, \quad \frac{48}{24} = 2, \quad \frac{50}{25} = 2 \][/tex]
All ratios are equal (2), so this triangle is similar to the 7, 24, 25 triangle.
- Answer: Yes
2. Set: 9, 12, 15
Let's check the ratios:
[tex]\[ \frac{9}{7} \approx 1.29, \quad \frac{12}{24} = 0.5, \quad \frac{15}{25} = 0.6 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
3. Set: [tex]\(2, \sqrt{20}, 2 \sqrt{6}\)[/tex]
Let's check the ratios:
[tex]\[ \frac{2}{7} \approx 0.29, \quad \frac{\sqrt{20}}{24} \approx 0.21, \quad \frac{2\sqrt{6}}{25} \approx 0.31 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
4. Set: 8, 15, 17
Let's check the ratios:
[tex]\[ \frac{8}{7} \approx 1.14, \quad \frac{15}{24} = 0.625, \quad \frac{17}{25} = 0.68 \][/tex]
The ratios are not equal, so this triangle is not similar.
- Answer: No
5. Set: [tex]\(\sqrt{7}, \sqrt{24}, \sqrt{25}\)[/tex]
Let's check the ratios:
[tex]\[ \frac{\sqrt{7}}{7}, \quad \frac{\sqrt{24}}{24}, \quad \frac{\sqrt{25}}{25} \][/tex]
[tex]\(\frac{\sqrt{7}}{7}\)[/tex] and [tex]\(\frac{\sqrt{24}}{24}: \sqrt{24} \approx 4.9, \frac{4.9}{24} \neq \frac{7}{7}\)[/tex]
[tex]\(\frac{\sqrt{25}}{25}: \frac{5}{25} \neq 1\)[/tex]
These ratios are not equal, so this triangle is not similar.
- Answer: No
6. Set: 35, 120, 125
Let's check the ratios:
[tex]\[ \frac{35}{7} = 5, \quad \frac{120}{24} = 5, \quad \frac{125}{25} = 5 \][/tex]
All ratios are equal (5), so this triangle is similar to the 7, 24, 25 triangle.
- Answer: Yes
7. Set: 21, 72, 78
Let's check the ratios:
[tex]\[ \frac{21}{7} = 3, \quad \frac{72}{24} = 3, \quad \frac{78}{25} \neq 3 \][/tex]
The ratios are not all equal, so this triangle is not similar (However, consideration may lead to a Yes answer from examining ratios - given constraints).
- Answer: No
After completing the analysis for each set, we conclude the sets of side lengths that form triangles similar to the triangle with side lengths 7, 24, and 25 are:
- [tex]\(14, 48, 50\)[/tex]
- \(35, 120, 125)
So, the correct answers are:
[tex]\[ \boxed{14, 48, 50 \text{ and } 35, 120, 125} \][/tex]
