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To determine which of the given sets of side lengths form triangles similar to a triangle with side lengths [tex]\(7, 24, 25\)[/tex], we need to check if the ratios of the corresponding sides in each set are the same as the ratios in the given Pythagorean triple.
### Explanation:
Triangles are similar if the corresponding sides are proportional. For the given Pythagorean triple [tex]\( (7, 24, 25) \)[/tex], let's denote the sides as [tex]\( (a_1, b_1, c_1) = (7, 24, 25) \)[/tex].
We need to check each of the given sets below to see if:
[tex]\( \frac{a_2}{a_1} = \frac{b_2}{b_1} = \frac{c_2}{c_1} \)[/tex]
where [tex]\( (a_2, b_2, c_2) \)[/tex] are the side lengths of the given sets.
### Check Each Set:
1. Set: [tex]\( (14, 48, 50) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{14}{7} = 2, \quad \frac{48}{24} = 2, \quad \frac{50}{25} = 2 \][/tex]
- All ratios are equal ([tex]\(2\)[/tex]), so this set forms a similar triangle.
2. Set: [tex]\( (9, 12, 15) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{9}{7}, \quad \frac{12}{24} = \frac{1}{2}, \quad \frac{15}{25} = \frac{3}{5} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
3. Set: [tex]\( (2, \sqrt{20}, 2 \sqrt{6}) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{2}{7}, \quad \frac{\sqrt{20}}{24}, \quad \frac{2 \sqrt{6}}{25} \][/tex]
- Given the irrational numbers, precise comparison shows the ratios are not equal, so this set does not form a similar triangle.
4. Set: [tex]\( (8, 15, 17) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{8}{7}, \quad \frac{15}{24}, \quad \frac{17}{25} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
5. Set: [tex]\( (\sqrt{7}, \sqrt{24}, \sqrt{25}) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{\sqrt{7}}{7}, \quad \frac{\sqrt{24}}{24}, \quad \frac{\sqrt{25}}{25} = \frac{5}{25} = \frac{1}{5} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
6. Set: [tex]\( (35, 120, 125) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{35}{7} \approx 5, \quad \frac{120}{24} = 5, \quad \frac{125}{25} = 5 \][/tex]
- All ratios are equal ([tex]\(5\)[/tex]), so this set forms a similar triangle.
### Conclusion
The sets that form triangles similar to a triangle with side lengths [tex]\( 7, 24, 25 \)[/tex] are:
[tex]\[ (14, 48, 50) \][/tex]
[tex]\[ (35, 120, 125) \][/tex]
### Explanation:
Triangles are similar if the corresponding sides are proportional. For the given Pythagorean triple [tex]\( (7, 24, 25) \)[/tex], let's denote the sides as [tex]\( (a_1, b_1, c_1) = (7, 24, 25) \)[/tex].
We need to check each of the given sets below to see if:
[tex]\( \frac{a_2}{a_1} = \frac{b_2}{b_1} = \frac{c_2}{c_1} \)[/tex]
where [tex]\( (a_2, b_2, c_2) \)[/tex] are the side lengths of the given sets.
### Check Each Set:
1. Set: [tex]\( (14, 48, 50) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{14}{7} = 2, \quad \frac{48}{24} = 2, \quad \frac{50}{25} = 2 \][/tex]
- All ratios are equal ([tex]\(2\)[/tex]), so this set forms a similar triangle.
2. Set: [tex]\( (9, 12, 15) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{9}{7}, \quad \frac{12}{24} = \frac{1}{2}, \quad \frac{15}{25} = \frac{3}{5} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
3. Set: [tex]\( (2, \sqrt{20}, 2 \sqrt{6}) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{2}{7}, \quad \frac{\sqrt{20}}{24}, \quad \frac{2 \sqrt{6}}{25} \][/tex]
- Given the irrational numbers, precise comparison shows the ratios are not equal, so this set does not form a similar triangle.
4. Set: [tex]\( (8, 15, 17) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{8}{7}, \quad \frac{15}{24}, \quad \frac{17}{25} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
5. Set: [tex]\( (\sqrt{7}, \sqrt{24}, \sqrt{25}) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{\sqrt{7}}{7}, \quad \frac{\sqrt{24}}{24}, \quad \frac{\sqrt{25}}{25} = \frac{5}{25} = \frac{1}{5} \][/tex]
- Ratios are not equal, so this set does not form a similar triangle.
6. Set: [tex]\( (35, 120, 125) \)[/tex]
- Calculate ratios:
[tex]\[ \frac{35}{7} \approx 5, \quad \frac{120}{24} = 5, \quad \frac{125}{25} = 5 \][/tex]
- All ratios are equal ([tex]\(5\)[/tex]), so this set forms a similar triangle.
### Conclusion
The sets that form triangles similar to a triangle with side lengths [tex]\( 7, 24, 25 \)[/tex] are:
[tex]\[ (14, 48, 50) \][/tex]
[tex]\[ (35, 120, 125) \][/tex]
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