Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Let's determine if each set of numbers can form the sides of a right triangle using the Pythagorean theorem. According to the Pythagorean theorem, for sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the relationship must satisfy [tex]\(a^2 + b^2 = c^2\)[/tex].
1. For [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]\(169 = 169\)[/tex], these numbers can form a right triangle.
- Answer: Yes
2. For [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]\(1369 \neq 1200\)[/tex], these numbers cannot form a right triangle.
- Answer: No
3. For [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]\(125 = 125\)[/tex], these numbers can form a right triangle.
- Answer: Yes
4. For [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]\(208 \neq 225\)[/tex], these numbers cannot form a right triangle.
- Answer: No
5. For [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]\(10201 = 10201\)[/tex], these numbers can form a right triangle.
- Answer: Yes
Thus, we fill out the table as follows:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$a$[/tex] & [tex]$b$[/tex] & [tex]$c$[/tex] & Pythagorean triple? \\
\hline
5 & 12 & 13 & Yes \\
\hline
12 & 35 & [tex]$20 \sqrt{3}$[/tex] & No \\
\hline
5 & 10 & [tex]$5 \sqrt{5}$[/tex] & Yes \\
\hline
8 & 12 & 15 & No \\
\hline
20 & 99 & 101 & Yes \\
\hline
\end{tabular}
1. For [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]\(169 = 169\)[/tex], these numbers can form a right triangle.
- Answer: Yes
2. For [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]\(1369 \neq 1200\)[/tex], these numbers cannot form a right triangle.
- Answer: No
3. For [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]\(125 = 125\)[/tex], these numbers can form a right triangle.
- Answer: Yes
4. For [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]\(208 \neq 225\)[/tex], these numbers cannot form a right triangle.
- Answer: No
5. For [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]\(10201 = 10201\)[/tex], these numbers can form a right triangle.
- Answer: Yes
Thus, we fill out the table as follows:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$a$[/tex] & [tex]$b$[/tex] & [tex]$c$[/tex] & Pythagorean triple? \\
\hline
5 & 12 & 13 & Yes \\
\hline
12 & 35 & [tex]$20 \sqrt{3}$[/tex] & No \\
\hline
5 & 10 & [tex]$5 \sqrt{5}$[/tex] & Yes \\
\hline
8 & 12 & 15 & No \\
\hline
20 & 99 & 101 & Yes \\
\hline
\end{tabular}
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
