301 Moved Permanently - Free Printable

Problem Overview:

We are tasked with determining whether the given sets of side lengths form a right triangle. A right triangle satisfies the Pythagorean theorem, which states:

\[
a^2 + b^2 = c^2
\]

where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse (the longest side).

Step-by-Step Solution:

#### Part 1: Check if the three sides form a right triangle

For each set of side lengths, we will:
1. Identify the longest side as the hypotenuse (\(c\)).
2. Verify if the equation \(a^2 + b^2 = c^2\) holds true.

---

#### Problem 1: 3, 4, 5
- Hypotenuse: \(c = 5\)
- Legs: \(a = 3\), \(b = 4\)
- Check: \(3^2 + 4^2 = 5^2\)
\[
9 + 16 = 25 \quad \text{(True)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 2: 6, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 6\), \(b = 8\)
- Check: \(6^2 + 8^2 = 10^2\)
\[
36 + 64 = 100 \quad \text{(True)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 3: 9, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 8\), \(b = 9\)
- Check: \(8^2 + 9^2 = 10^2\)
\[
64 + 81 = 145 \quad \text{(False, since } 145 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 4: 5, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 5\), \(b = 8\)
- Check: \(5^2 + 8^2 = 10^2\)
\[
25 + 64 = 89 \quad \text{(False, since } 89 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 5: 6, 9, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 6\), \(b = 9\)
- Check: \(6^2 + 9^2 = 10^2\)
\[
36 + 81 = 117 \quad \text{(False, since } 117 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 6: 6, 6, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 6\), \(b = 6\)
- Check: \(6^2 + 6^2 = 10^2\)
\[
36 + 36 = 72 \quad \text{(False, since } 72 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 7: 7, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 7\), \(b = 8\)
- Check: \(7^2 + 8^2 = 10^2\)
\[
49 + 64 = 113 \quad \text{(False, since } 113 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 8: 4, 4, 5
- Hypotenuse: \(c = 5\)
- Legs: \(a = 4\), \(b = 4\)
- Check: \(4^2 + 4^2 = 5^2\)
\[
16 + 16 = 32 \quad \text{(False, since } 32 \neq 25\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 9: 3, 3, 5
- Hypotenuse: \(c = 5\)
- Legs: \(a = 3\), \(b = 3\)
- Check: \(3^2 + 3^2 = 5^2\)
\[
9 + 9 = 18 \quad \text{(False, since } 18 \neq 25\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 10: 6, 8, 9
- Hypotenuse: \(c = 9\)
- Legs: \(a = 6\), \(b = 8\)
- Check: \(6^2 + 8^2 = 9^2\)
\[
36 + 64 = 100 \quad \text{(False, since } 100 \neq 81\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 11: 11, 60, 61
- Hypotenuse: \(c = 61\)
- Legs: \(a = 11\), \(b = 60\)
- Check: \(11^2 + 60^2 = 61^2\)
\[
121 + 3600 = 3721 \quad \text{(True, since } 3721 = 3721\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 12: 42, 56, 70
- Hypotenuse: \(c = 70\)
- Legs: \(a = 42\), \(b = 56\)
- Check: \(42^2 + 56^2 = 70^2\)
\[
1764 + 3136 = 4900 \quad \text{(True, since } 4900 = 4900\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 13: 42, 56, 75
- Hypotenuse: \(c = 75\)
- Legs: \(a = 42\), \(b = 56\)
- Check: \(42^2 + 56^2 = 75^2\)
\[
1764 + 3136 = 4900 \quad \text{(False, since } 4900 \neq 5625\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 14: 24, 70, 76
- Hypotenuse: \(c = 76\)
- Legs: \(a = 24\), \(b = 70\)
- Check: \(24^2 + 70^2 = 76^2\)
\[
576 + 4900 = 5476 \quad \text{(True, since } 5476 = 5476\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 15: 48, 55, 73
- Hypotenuse: \(c = 73\)
- Legs: \(a = 48\), \(b = 55\)
- Check: \(48^2 + 55^2 = 73^2\)
\[
2304 + 3025 = 5329 \quad \text{(True, since } 5329 = 5329\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 16: 57, 76, 95
- Hypotenuse: \(c = 95\)
- Legs: \(a = 57\), \(b = 76\)
- Check: \(57^2 + 76^2 = 95^2\)
\[
3249 + 5776 = 9025 \quad \text{(True, since } 9025 = 9025\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 17: 47, 55, 73
- Hypotenuse: \(c = 73\)
- Legs: \(a = 47\), \(b = 55\)
- Check: \(47^2 + 55^2 = 73^2\)
\[
2209 + 3025 = 5234 \quad \text{(False, since } 5234 \neq 5329\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 18: 24, 73, 74
- Hypotenuse: \(c = 74\)
- Legs: \(a = 24\), \(b = 73\)
- Check: \(24^2 + 73^2 = 74^2\)
\[
576 + 5329 = 5905 \quad \text{(False, since } 5905 \neq 5476\text{)}
\]
- Conclusion: This is not a right triangle.

---

#### Problem 19: 48, 64, 80
- Hypotenuse: \(c = 80\)
- Legs: \(a = 48\), \(b = 64\)
- Check: \(48^2 + 64^2 = 80^2\)
\[
2304 + 4096 = 6400 \quad \text{(True, since } 6400 = 6400\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Problem 20: 13, 84, 89
- Hypotenuse: \(c = 89\)
- Legs: \(a = 13\), \(b = 84\)
- Check: \(13^2 + 84^2 = 89^2\)
\[
169 + 7056 = 7225 \quad \text{(True, since } 7225 = 7225\text{)}
\]
- Conclusion: This is a right triangle.

---

#### Part 2: Check if each triangle is a right triangle

##### Problem 21: Triangle with sides 8, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 8\), \(b = 8\)
- Check: \(8^2 + 8^2 = 10^2\)
\[
64 + 64 = 128 \quad \text{(False, since } 128 \neq 100\text{)}
\]
- Conclusion: This is not a right triangle.

##### Problem 22: Triangle with sides 6, 8, 10
- Hypotenuse: \(c = 10\)
- Legs: \(a = 6\), \(b = 8\)
- Check: \(6^2 + 8^2 = 10^2\)
\[
36 + 64 = 100 \quad \text{(True)}
\]
- Conclusion: This is a right triangle.

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) \text{Yes} & 2) \text{Yes} \\
3) \text{No} & 4) \text{No} \\
5) \text{No} & 6) \text{No} \\
7) \text{No} & 8) \text{No} \\
9) \text{No} & 10) \text{No} \\
11) \text{Yes} & 12) \text{Yes} \\
13) \text{No} & 14) \text{Yes} \\
15) \text{Yes} & 16) \text{Yes} \\
17) \text{No} & 18) \text{No} \\
19) \text{Yes} & 20) \text{Yes} \\
21) \text{No} & 22) \text{Yes} \\
\end{array}
}
\]