The problem requires identifying which sets of numbers form triangles similar to a triangle with side lengths [tex]\(7, 24, 25\)[/tex]. To determine similarity, you must check if the given sets have the same ratios as the side lengths of the original triangle. Here’s a step-by-step solution:
1. Understanding the Concept of Similarity:
Two triangles are similar if their corresponding sides are in the same ratio. For the given triangle with sides [tex]\(7, 24, 25\)[/tex]:
[tex]\[ \text{Ratio of sides: } \left(\frac{7}{a}, \frac{24}{b}, \frac{25}{c}\right) \][/tex]
2. Given Sets of Side Lengths:
- [tex]\( 14, 48, 50 \)[/tex]
- [tex]\( 9, 12, 15 \)[/tex]
- [tex]\( 2, \sqrt{20}, 2\sqrt{6} \)[/tex]
- [tex]\( 8, 15, 17 \)[/tex]
- [tex]\( \sqrt{7}, \sqrt{24}, \sqrt{25} \)[/tex]
- [tex]\( 35, 120, 125 \)[/tex]
- [tex]\( 21, 72, 78 \)[/tex]
3. Checking Each Set:
- [tex]\( 14, 48, 50 \)[/tex]:
[tex]\[ \frac{14}{7} = 2, \ \frac{48}{24} = 2, \ \frac{50}{25} = 2 \][/tex]
Ratios are the same, so this set is similar.
- [tex]\( 9, 12, 15 \)[/tex]:
[tex]\[ \frac{9}{7}, \frac{12}{24} = 0.5, \frac{15}{25} = 0.6 \][/tex]
Ratios are not the same, so this set is not similar.
- [tex]\( 2, \sqrt{20}, 2\sqrt{6} \)[/tex]:
[tex]\[ \frac{2}{7}, \frac{\sqrt{20}}{24}, \frac{2\sqrt{6}}{25} \][/tex]
[tex]\( \sqrt{20} \approx 4.47 \)[/tex]
[tex]\( \frac{\sqrt{20}}{24} \approx 0.186 \neq \frac{2}{7} \approx 0.286 \)[/tex]
Ratios are not the same, so this set is not similar.
- [tex]\( 8, 15, 17 \)[/tex]:
[tex]\[ \frac{8}{7}, \frac{15}{24}, \frac{17}{25} \][/tex]
Ratios are not the same, so this set is not similar.
- [tex]\( \sqrt{7}, \sqrt{24}, \sqrt{25} \)[/tex]:
[tex]\[ \frac{\sqrt{7}}{7}, \frac{\sqrt{24}}{24}, \frac{\sqrt{25}}{25} \][/tex]
Ratios do not match due to the square roots, so this set is not similar.
- [tex]\( 35, 120, 125 \)[/tex]:
[tex]\[ \frac{35}{7} = 5, \ \frac{120}{24} = 5, \ \frac{125}{25} = 5 \][/tex]
Ratios are the same, so this set is similar.
- [tex]\( 21, 72, 78 \)[/tex]:
[tex]\[ \frac{21}{7} = 3, \ \frac{72}{24} = 3, \ \frac{78}{25} = 3.12 \][/tex]
Ratios are not the same, so this set is not similar.
4. Conclusion:
The sets that form triangles similar to a triangle with side lengths [tex]\(7, 24, 25\)[/tex] are:
[tex]\[ \boxed{14, 48, 50 \text{ and } 35, 120, 125} \][/tex]
