To determine which of the given sets are the side lengths of triangles similar to a triangle whose side lengths measure [tex]\(7, 24, 25\)[/tex], we need to check if each set of side lengths maintains the same ratio as the given Pythagorean triple. Two triangles are similar if their corresponding side lengths are proportional.
The sets provided are:
1. [tex]\( (14, 48, 50) \)[/tex]
2. [tex]\( (21, 72, 75) \)[/tex]
3. [tex]\( (28, 96, 100) \)[/tex]
4. [tex]\( (35, 120, 125) \)[/tex]
We will verify each set individually by comparing the ratios.
### Step-by-step Solution:
1. Given Pythagorean triple: [tex]\( (7, 24, 25) \)[/tex]
- Here, the sides [tex]\(a = 7\)[/tex], [tex]\(b = 24\)[/tex], and [tex]\(c = 25\)[/tex].
The ratio of the sides of this triangle (using [tex]\(a\)[/tex] to [tex]\(c\)[/tex]) is:
[tex]\[
\frac{c}{a} = \frac{25}{7}
\][/tex]
2. Checking each given set:
- Set (14, 48, 50):
- [tex]\(a = 14\)[/tex], [tex]\(b = 48\)[/tex], [tex]\(c = 50\)[/tex]
- Ratio:
[tex]\[
\frac{c}{a} = \frac{50}{14} = \frac{25}{7}
\][/tex]
This matches the ratio of the given Pythagorean triple.
- Set (21, 72, 75):
- [tex]\(a = 21\)[/tex], [tex]\(b = 72\)[/tex], [tex]\(c = 75\)[/tex]
- Ratio:
[tex]\[
\frac{c}{a} = \frac{75}{21} = \frac{25}{7}
\][/tex]
This matches the ratio of the given Pythagorean triple.
- Set (28, 96, 100):
- [tex]\(a = 28\)[/tex], [tex]\(b = 96\)[/tex], [tex]\(c = 100\)[/tex]
- Ratio:
[tex]\[
\frac{c}{a} = \frac{100}{28} = \frac{25}{7}
\][/tex]
This matches the ratio of the given Pythagorean triple.
- Set (35, 120, 125):
- [tex]\(a = 35\)[/tex], [tex]\(b = 120\)[/tex], [tex]\(c = 125\)[/tex]
- Ratio:
[tex]\[
\frac{c}{a} = \frac{125}{35} = \frac{25}{7}
\][/tex]
This matches the ratio of the given Pythagorean triple.
Therefore, all the provided sets are in the same ratio as the given Pythagorean triple, which means all of them represent triangles similar to the triangle with side lengths [tex]\(7, 24, 25\)[/tex].
### Final Answer
The correct sets are:
- [tex]\( (14, 48, 50) \)[/tex]
- [tex]\( (21, 72, 75) \)[/tex]
- [tex]\( (28, 96, 100) \)[/tex]
- [tex]\( (35, 120, 125) \)[/tex]
