To classify a triangle as acute, right, or obtuse based on its side lengths, we first need to determine the type of triangle using the longest side.
Given the side lengths: 10, 49.5, and 50.5, we can identify the longest side as 50.5. We can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
- Check the sums:
- 10 + 49.5 = 59.5, which is greater than 50.5 (valid)
- 10 + 50.5 = 60.5, which is greater than 49.5 (valid)
- 49.5 + 50.5 = 100, which is greater than 10 (valid)
Since all conditions are satisfied, these side lengths form a valid triangle.
- To classify the triangle:
- We can use the relationship based on the square of the side lengths.
Let \( a = 10 \), \( b = 49.5 \), and \( c = 50.5 \):
- Calculate \( a^2 + b^2 \) and \( c^2 \):
- \( a^2 = 10^2 = 100 \)
- \( b^2 = 49.5^2 \approx 2450.25 \)
- \( c^2 = 50.5^2 \approx 2550.25 \)
Now compare:
- If \( a^2 + b^2 < c^2 \), the triangle is obtuse.
- If \( a^2 + b^2 = c^2 \), the triangle is right.
- If \( a^2 + b^2 > c^2 \), the triangle is acute.
Now calculate:
- \( a^2 + b^2 \approx 100 + 2450.25 = 2550.25 \)
- Since \( a^2 + b^2 = c^2 \) (approximately, since they are very close), we can classify this triangle as right.
Therefore, the triangle with side lengths 10, 49.5, and 50.5 forms a right triangle.