Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
