To determine if each set of numbers can be the lengths of the sides of a right triangle, we need to check if they satisfy the Pythagorean theorem, which states that for a right triangle with legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and hypotenuse [tex]\(c\)[/tex],
[tex]\[ a^2 + b^2 = c^2. \][/tex]
Let's examine each case:
1. Set 1: [tex]\(a = 5\)[/tex], [tex]\(b = 12\)[/tex], [tex]\(c = 13\)[/tex]
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[ 5^2 + 12^2 = 25 + 144 = 169. \][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[ 13^2 = 169. \][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex] holds true, this is a Pythagorean triple.
Answer: Yes
2. Set 2: [tex]\(a = 12\)[/tex], [tex]\(b = 35\)[/tex], [tex]\(c = 20 \sqrt{3}\)[/tex]
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[ 12^2 + 35^2 = 144 + 1225 = 1369. \][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[ (20 \sqrt{3})^2 = 400 \times 3 = 1200. \][/tex]
- Since [tex]\(a^2 + b^2 \neq c^2\)[/tex], this is not a Pythagorean triple.
Answer: No
3. Set 3: [tex]\(a = 5\)[/tex], [tex]\(b = 10\)[/tex], [tex]\(c = 5 \sqrt{5}\)[/tex]
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[ 5^2 + 10^2 = 25 + 100 = 125. \][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[ (5 \sqrt{5})^2 = 25 \times 5 = 125. \][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex] holds true, this is a Pythagorean triple.
Answer: Yes
4. Set 4: [tex]\(a = 8\)[/tex], [tex]\(b = 12\)[/tex], [tex]\(c = 15\)[/tex]
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[ 8^2 + 12^2 = 64 + 144 = 208. \][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[ 15^2 = 225. \][/tex]
- Since [tex]\(a^2 + b^2 \neq c^2\)[/tex], this is not a Pythagorean triple.
Answer: No
5. Set 5: [tex]\(a = 20\)[/tex], [tex]\(b = 99\)[/tex], [tex]\(c = 101\)[/tex]
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[ 20^2 + 99^2 = 400 + 9801 = 10201. \][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[ 101^2 = 10201. \][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex] holds true, this is a Pythagorean triple.
Answer: Yes
Based on the verification, the table should be filled as follows:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
a & b & c & \text{Pythagorean triple?} \\
\hline
5 & 12 & 13 & \text{Yes} \\
\hline
12 & 35 & 20 \sqrt{3} & \text{No} \\
\hline
5 & 10 & 5 \sqrt{5} & \text{Yes} \\
\hline
8 & 12 & 15 & \text{No} \\
\hline
20 & 99 & 101 & \text{Yes} \\
\hline
\end{tabular}
\][/tex]
