To determine which of the given sets of side lengths represent a right triangle, we will use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
Let's analyze each set of triangle side lengths provided.
### Step 1: Identify the sets of side lengths
Assuming the side lengths are:
1. \( \sqrt{5}, 2, \sqrt{7} \)
2. \( 1, 4, 5 \)
3. \( \sqrt{3}, \sqrt{4}, \sqrt{5} \)
4. \( 3, 4, 5 \)
5. \( 2, 2, 3 \)
### Step 2: Check each set against the Pythagorean theorem
#### Set 1: \( \sqrt{5}, 2, \sqrt{7} \)
- Identify the longest side: \( \sqrt{7} \)
- Check:
\[
\sqrt{7}^2 \stackrel{?}{=} \sqrt{5}^2 + 2^2
\]
\[
7 \stackrel{?}{=} 5 + 4 \Rightarrow 7 = 9 \quad \text{(False)}
\]
#### Set 2: \( 1, 4, 5 \)
- Identify the longest side: \( 5 \)
- Check:
\[
5^2 \stackrel{?}{=} 1^2 + 4^2
\]
\[
25 \stackrel{?}{=} 1 + 16 \Rightarrow 25 = 17 \quad \text{(False)}
\]
#### Set 3: \( \sqrt{3}, \sqrt{4}, \sqrt{5} \)
- Identify the longest side: \( \sqrt{5} \)
- Check:
\[
\sqrt{5}^2 \stackrel{?}{=} \sqrt{3}^2 + \sqrt{4}^2
\]
\[
5 \stackrel{?}{=} 3 + 4 \Rightarrow 5 = 7 \quad \text{(False)}
\]
#### Set 4: \( 3, 4, 5 \)
- Identify the longest side: \( 5 \)
- Check:
\[
5^2 \stackrel{?}{=} 3^2 + 4^2
\]
\[
25 \stackrel{?}{=} 9 + 16 \Rightarrow 25 = 25 \quad \text{(True)}
\]
#### Set 5: \( 2, 2, 3 \)
- Identify the longest side: \( 3 \)
- Check:
\[
3^2 \stackrel{?}{=} 2^2 + 2^2
\]
\[
9 \stackrel{?}{=} 4 + 4 \Rightarrow 9 = 8 \quad \text{(False)}
\]
### Conclusion
The only set of side lengths that satisfies the Pythagorean theorem is \( 3, 4, 5 \). Therefore, this set represents a right triangle.
### Final Answer
The set of sides that represents a right triangle is \( 3, 4, 5 \).
