To determine which set of numbers can represent the side lengths of a right triangle, we use the Pythagorean Theorem. According to the theorem, 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. This means for side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\(c\)[/tex] is the hypotenuse:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's evaluate each set of numbers to see if they satisfy this condition:
1. Set: [tex]\(8, 12, 15\)[/tex]
We need to check if:
[tex]\[ 8^2 + 12^2 = 15^2 \][/tex]
[tex]\[
8^2 = 64 \\
12^2 = 144 \\
15^2 = 225
\][/tex]
Therefore:
[tex]\[ 64 + 144 = 208 \][/tex]
[tex]\[ 208 \neq 225 \][/tex]
This set does not represent a right triangle.
2. Set: [tex]\(10, 24, 26\)[/tex]
We need to check if:
[tex]\[ 10^2 + 24^2 = 26^2 \][/tex]
[tex]\[
10^2 = 100 \\
24^2 = 576 \\
26^2 = 676
\][/tex]
Therefore:
[tex]\[ 100 + 576 = 676 \][/tex]
[tex]\[ 676 = 676 \][/tex]
This set represents a right triangle.
3. Set: [tex]\(12, 20, 25\)[/tex]
We need to check if:
[tex]\[ 12^2 + 20^2 = 25^2 \][/tex]
[tex]\[
12^2 = 144 \\
20^2 = 400 \\
25^2 = 625
\][/tex]
Therefore:
[tex]\[ 144 + 400 = 544 \][/tex]
[tex]\[ 544 \neq 625 \][/tex]
This set does not represent a right triangle.
4. Set: [tex]\(15, 18, 20\)[/tex]
We need to check if:
[tex]\[ 15^2 + 18^2 = 20^2 \][/tex]
[tex]\[
15^2 = 225 \\
18^2 = 324 \\
20^2 = 400
\][/tex]
Therefore:
[tex]\[ 225 + 324 = 549 \][/tex]
[tex]\[ 549 \neq 400 \][/tex]
This set does not represent a right triangle.
Therefore, the set of numbers that can represent the side lengths, in centimeters, of a right triangle is:
[tex]\[
(10, 24, 26)
\][/tex]
