To determine if a set of numbers [tex]\((a, b, c)\)[/tex] can be the lengths of the sides of a right triangle, we use the Pythagorean Theorem. This theorem states that for any right triangle with legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the following equation must hold true:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's analyze each set in the table:
1. Set (5, 12, 13):
To check if these numbers form a right triangle, calculate:
[tex]\[
5^2 + 12^2 = 25 + 144 = 169
\][/tex]
[tex]\[
13^2 = 169
\][/tex]
Since [tex]\( 5^2 + 12^2 = 13^2 \)[/tex], the numbers 5, 12, and 13 can form the sides of a right triangle.
Answer: Yes
2. Set (12, 35, [tex]\(20 \sqrt{3}\)[/tex]):
Calculate:
[tex]\[
12^2 + 35^2 = 144 + 1225 = 1369
\][/tex]
[tex]\[
(20 \sqrt{3})^2 = 400 \cdot 3 = 1200
\][/tex]
Since [tex]\( 12^2 + 35^2 \neq (20 \sqrt{3})^2 \)[/tex], the numbers 12, 35, and [tex]\(20 \sqrt{3}\)[/tex] cannot form the sides of a right triangle.
Answer: No
3. Set (5, 10, [tex]\(5 \sqrt{5}\)[/tex]):
Calculate:
[tex]\[
5^2 + 10^2 = 25 + 100 = 125
\][/tex]
[tex]\[
(5 \sqrt{5})^2 = 25 \cdot 5 = 125
\][/tex]
Since [tex]\( 5^2 + 10^2 = (5 \sqrt{5})^2 \)[/tex], the numbers 5, 10, and [tex]\(5 \sqrt{5}\)[/tex] can form the sides of a right triangle.
Answer: Yes
4. Set (8, 12, 15):
Calculate:
[tex]\[
8^2 + 12^2 = 64 + 144 = 208
\][/tex]
[tex]\[
15^2 = 225
\][/tex]
Since [tex]\( 8^2 + 12^2 \neq 15^2 \)[/tex], the numbers 8, 12, and 15 cannot form the sides of a right triangle.
Answer: No
5. Set (20, 99, 101):
Calculate:
[tex]\[
20^2 + 99^2 = 400 + 9801 = 10201
\][/tex]
[tex]\[
101^2 = 10201
\][/tex]
Since [tex]\( 20^2 + 99^2 = 101^2 \)[/tex], the numbers 20, 99, and 101 can form the sides of a right triangle.
Answer: Yes
Based on the calculations, the correct table will be:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$\boldsymbol{a}$ & $\boldsymbol{b}$ & $\boldsymbol{c}$ & \textbf{Pythagorean triple} \\
\hline
5 & 12 & 13 & Yes \\
\hline
12 & 35 & $20 \sqrt{3}$ & No \\
\hline
5 & 10 & $5 \sqrt{5}$ & No \\
\hline
8 & 12 & 15 & No \\
\hline
20 & 99 & 101 & Yes \\
\hline
\end{tabular}
\][/tex]
