To classify the triangle with side lengths 4.8, 28.6, and 29, we first need to check if it satisfies the triangle inequality theorem:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let’s assign \( a = 4.8 \), \( b = 28.6 \), and \( c = 29 \).
- \( 4.8 + 28.6 = 33.4 > 29 \) (True)
- \( 4.8 + 29 = 33.8 > 28.6 \) (True)
- \( 28.6 + 29 = 57.6 > 4.8 \) (True)
Since all inequalities are satisfied, these sides can form a triangle.
Next, to classify the triangle (acute, right, or obtuse) based on the angles, we will use the converse of the Pythagorean theorem. We will compare the square of the longest side (\( c = 29 \)) with the sum of the squares of the other two sides (\( a \) and \( b \)).
Calculate \( a^2 + b^2 \): \[ (4.8)^2 + (28.6)^2 = 23.04 + 817.96 = 841 \]
Calculate \( c^2 \): \[ (29)^2 = 841 \]
Now we compare \( a^2 + b^2 \) and \( c^2 \): \[ a^2 + b^2 = 841 \quad \text{and} \quad c^2 = 841 \]
Since \( a^2 + b^2 = c^2 \), the triangle is a right triangle.
Conclusion: The triangle with sides 4.8, 28.6, and 29 is a right triangle.