Answer :
To determine which of the given sets of side lengths form triangles that are similar to the triangle with side lengths 7, 24, and 25, we need to check two criteria:
1. They must form a Pythagorean triple.
2. The triangles must be similar, meaning the side lengths of the given sets must be proportional to 7, 24, and 25.
### Step-by-Step Solution
#### Given Triangle
The original triangle has side lengths:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 24 \)[/tex]
- [tex]\( c = 25 \)[/tex]
It forms a Pythagorean triple, as [tex]\(7^2 + 24^2 = 25^2\)[/tex].
#### Checking Each Set
1. Set: [tex]\( 14, 48, 50 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 14^2 + 48^2 = 196 + 2304 = 2500 = 50^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{14}{7} = 2 \)[/tex], [tex]\( \frac{48}{24} = 2 \)[/tex], [tex]\( \frac{50}{25} = 2 \)[/tex]
- This set meets both criteria and is similar to the original triangle.
2. Set: [tex]\( 9, 12, 15 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 9^2 + 12^2 = 81 + 144 = 225 = 15^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{9}{7} \neq \frac{12}{24} \)[/tex], hence not the same ratio for all sides.
- This set does not meet the similarity criteria.
3. Set: [tex]\( 2, \sqrt{20}, 2\sqrt{6} \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 2^2 + (\sqrt{20})^2 = 4 + 20 = 24 \neq (2\sqrt{6})^2 = 24 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
4. Set: [tex]\( 8, 15, 17 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 8^2 + 15^2 = 64 + 225 = 289 = 17^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{8}{7}, \frac{15}{24}, \frac{17}{25} \)[/tex] are not in the same ratio.
- This set does not meet the similarity criteria.
5. Set: [tex]\( \sqrt{7}, \sqrt{24}, \sqrt{25} \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( (\sqrt{7})^2 + (\sqrt{24})^2 = 7 + 24 = 31 \neq (\sqrt{25})^2 = 25 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
6. Set: [tex]\( 35, 120, 125 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 35^2 + 120^2 = 1225 + 14400 = 15625 = 125^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{35}{7} = 5, \frac{120}{24} = 5, \frac{125}{25} = 5 \)[/tex]
- This set meets both criteria and is similar to the original triangle.
7. Set: [tex]\( 21, 72, 78 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 21^2 + 72^2 = 441 + 5184 = 5625 \neq 78^2 = 6084 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
### Conclusion
The sets that form side lengths of triangles similar to a triangle with side lengths 7, 24, and 25 are:
- [tex]\( 14, 48, 50 \)[/tex]
- [tex]\( 35, 120, 125 \)[/tex]
Therefore, the correct options are:
[tex]\[ \boxed{14,48,50} \][/tex]
[tex]\[ \boxed{35,120,125} \][/tex]
1. They must form a Pythagorean triple.
2. The triangles must be similar, meaning the side lengths of the given sets must be proportional to 7, 24, and 25.
### Step-by-Step Solution
#### Given Triangle
The original triangle has side lengths:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 24 \)[/tex]
- [tex]\( c = 25 \)[/tex]
It forms a Pythagorean triple, as [tex]\(7^2 + 24^2 = 25^2\)[/tex].
#### Checking Each Set
1. Set: [tex]\( 14, 48, 50 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 14^2 + 48^2 = 196 + 2304 = 2500 = 50^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{14}{7} = 2 \)[/tex], [tex]\( \frac{48}{24} = 2 \)[/tex], [tex]\( \frac{50}{25} = 2 \)[/tex]
- This set meets both criteria and is similar to the original triangle.
2. Set: [tex]\( 9, 12, 15 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 9^2 + 12^2 = 81 + 144 = 225 = 15^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{9}{7} \neq \frac{12}{24} \)[/tex], hence not the same ratio for all sides.
- This set does not meet the similarity criteria.
3. Set: [tex]\( 2, \sqrt{20}, 2\sqrt{6} \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 2^2 + (\sqrt{20})^2 = 4 + 20 = 24 \neq (2\sqrt{6})^2 = 24 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
4. Set: [tex]\( 8, 15, 17 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 8^2 + 15^2 = 64 + 225 = 289 = 17^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{8}{7}, \frac{15}{24}, \frac{17}{25} \)[/tex] are not in the same ratio.
- This set does not meet the similarity criteria.
5. Set: [tex]\( \sqrt{7}, \sqrt{24}, \sqrt{25} \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( (\sqrt{7})^2 + (\sqrt{24})^2 = 7 + 24 = 31 \neq (\sqrt{25})^2 = 25 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
6. Set: [tex]\( 35, 120, 125 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 35^2 + 120^2 = 1225 + 14400 = 15625 = 125^2 \)[/tex]
- Proportional to the original triangle: [tex]\( \frac{35}{7} = 5, \frac{120}{24} = 5, \frac{125}{25} = 5 \)[/tex]
- This set meets both criteria and is similar to the original triangle.
7. Set: [tex]\( 21, 72, 78 \)[/tex]
- Check if it forms a Pythagorean triple: [tex]\( 21^2 + 72^2 = 441 + 5184 = 5625 \neq 78^2 = 6084 \)[/tex]
- The sides are not even forming a Pythagorean triple.
- This set does not meet the criteria.
### Conclusion
The sets that form side lengths of triangles similar to a triangle with side lengths 7, 24, and 25 are:
- [tex]\( 14, 48, 50 \)[/tex]
- [tex]\( 35, 120, 125 \)[/tex]
Therefore, the correct options are:
[tex]\[ \boxed{14,48,50} \][/tex]
[tex]\[ \boxed{35,120,125} \][/tex]
