To determine which sets of side lengths form a right triangle, we use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. That is, if [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the side lengths of a right triangle with [tex]\(c\)[/tex] being the hypotenuse, then:
[tex]\[a^2 + b^2 = c^2\][/tex]
Let’s analyze each set of side lengths one by one.
### Set 1: [tex]\(3, 14, \sqrt{205}\)[/tex]
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 14\)[/tex]
- [tex]\(c = \sqrt{205}\)[/tex]
Check:
[tex]\[3^2 + 14^2 = 9 + 196 = 205\][/tex]
[tex]\((\sqrt{205})^2 = 205\]
Since \(205 = 205\)[/tex], this set of side lengths forms a right triangle.
### Set 2: [tex]\(6, 11, \sqrt{158}\)[/tex]
- [tex]\(a = 6\)[/tex]
- [tex]\(b = 11\)[/tex]
- [tex]\(c = \sqrt{158}\)[/tex]
Check:
[tex]\[6^2 + 11^2 = 36 + 121 = 157\][/tex]
[tex]\((\sqrt{158})^2 = 158\]
Since \(157 \neq 158\)[/tex], this set of side lengths does not form a right triangle.
### Set 3: [tex]\(19, 180, 181\)[/tex]
- [tex]\(a = 19\)[/tex]
- [tex]\(b = 180\)[/tex]
- [tex]\(c = 181\)[/tex]
Check:
[tex]\[19^2 + 180^2 = 361 + 32400 = 32761\][/tex]
[tex]\[181^2 = 32761\][/tex]
Since [tex]\(32761 = 32761\)[/tex], this set of side lengths forms a right triangle.
### Set 4: [tex]\(3, 19, \sqrt{380}\)[/tex]
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 19\)[/tex]
- [tex]\(c = \sqrt{380}\)[/tex]
Check:
[tex]\[3^2 + 19^2 = 9 + 361 = 370\][/tex]
[tex]\((\sqrt{380})^2 = 380\]
Since \(370 \neq 380\)[/tex], this set of side lengths does not form a right triangle.
### Set 5: [tex]\(2, 9, \sqrt{85}\)[/tex]
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = \sqrt{85}\)[/tex]
Check:
[tex]\[2^2 + 9^2 = 4 + 81 = 85\][/tex]
[tex]\((\sqrt{85})^2 = 85\]
Since \(85 = 85\)[/tex], this set of side lengths forms a right triangle.
### Summary
The sets of side lengths that form right triangles are:
- [tex]\(3, 14, \sqrt{205}\)[/tex]
- [tex]\(19, 180, 181\)[/tex]
- [tex]\(2, 9, \sqrt{85}\)[/tex]
