To determine which of the given sets of numbers could not represent the three sides of a right triangle, we need to use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex]:
[tex]\[a^2 + b^2 = c^2\][/tex]
We will test each set to see if this condition holds true.
1. Set {6, 8, 10}:
- Assuming [tex]\(a = 6\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 10\)[/tex]:
[tex]\[
a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100\\
c^2 = 10^2 = 100
\][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents a right triangle.
2. Set {25, 60, 65}:
- Assuming [tex]\(a = 25\)[/tex], [tex]\(b = 60\)[/tex], and [tex]\(c = 65\)[/tex]:
[tex]\[
a^2 + b^2 = 25^2 + 60^2 = 625 + 3600 = 4225\\
c^2 = 65^2 = 4225
\][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents a right triangle.
3. Set {30, 40, 50}:
- Assuming [tex]\(a = 30\)[/tex], [tex]\(b = 40\)[/tex], and [tex]\(c = 50\)[/tex]:
[tex]\[
a^2 + b^2 = 30^2 + 40^2 = 900 + 1600 = 2500\\
c^2 = 50^2 = 2500
\][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents a right triangle.
4. Set {13, 84, 86}:
- Assuming [tex]\(a = 13\)[/tex], [tex]\(b = 84\)[/tex], and [tex]\(c = 86\)[/tex]:
[tex]\[
a^2 + b^2 = 13^2 + 84^2 = 169 + 7056 = 7225\\
c^2 = 86^2 = 7396
\][/tex]
Since [tex]\(a^2 + b^2 \neq c^2\)[/tex], this set does not represent a right triangle.
In conclusion, the set of numbers [tex]\(\{13, 84, 86\}\)[/tex] could not represent the three sides of a right triangle.
