To determine the type of triangle formed by the given side lengths 12, 35, and 37, we can use the properties of triangles concerning the squares of their sides.
### Step-by-Step Solution:
1. Identify the sides:
- Side1: 12
- Side2: 35
- Side3: 37
2. Determine the longest side:
- The longest side among 12, 35, and 37 is 37.
3. Calculate the square of the longest side:
- Longest side squared: [tex]\(37^2 = 1369\)[/tex].
4. Identify the other two sides as the shorter sides:
- The other two sides are 12 and 35.
5. Calculate the sum of the squares of the other two sides:
- Sum of squares: [tex]\(12^2 + 35^2 = (144) + (1225) = 1369\)[/tex].
6. Compare the square of the longest side to the sum of the squares of the other two sides:
- We need to check the relationship:
- If [tex]\( \text{longest}^2 = \text{other1}^2 + \text{other2}^2 \)[/tex], the triangle is right.
- If [tex]\( \text{longest}^2 < \text{other1}^2 + \text{other2}^2 \)[/tex], the triangle is acute.
- If [tex]\( \text{longest}^2 > \text{other1}^2 + \text{other2}^2 \)[/tex], the triangle is obtuse.
7. Conclusion:
- Here, [tex]\( 37^2 = 12^2 + 35^2 \)[/tex].
- Therefore, [tex]\( 1369 = 1369 \)[/tex].
Because the square of the longest side is exactly equal to the sum of the squares of the other two sides:
### The triangle with side lengths 12, 35, and 37 is a right triangle.
