Classify the triangle as right, acute, or obtuse if the measures of its sides are 28, 46, and 53.

It is acute

Use the cosine rule:

cos(A)=(b²+c²-a²)/(2bc)

When a is the hypotenuse of a right-triangle, cos(A)=0.
When a<90° cos(A)>0
When a>90° cos(A)<0

So by evaluating the cosine opposite the longest side, you can tell if the triangle is a right-triangle, acute or obtuse.

To classify a triangle as right, acute, or obtuse, we need to analyze the relationship between the lengths of its sides.

In this case, the given side lengths are 28, 46, and 53.

To determine the type of triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's calculate the squares of the side lengths:
28^2 = 784
46^2 = 2116
53^2 = 2809

Now, let's check if the Pythagorean theorem is satisfied:
Is the sum of the squares of the two smaller sides equal to the square of the largest side?

784 + 2116 = 2900

Since 2900 is not equal to 2809, the Pythagorean theorem is not satisfied. This means that the triangle is not a right triangle.

To determine if the triangle is acute or obtuse, we can compare the squares of the side lengths. In an acute triangle, the sum of the squares of the two smaller sides is greater than the square of the largest side. In an obtuse triangle, the square of the largest side is greater than the sum of the squares of the other two sides.

In this case, the sum of the squares of the smaller sides is 784 + 2116 = 2900, which is less than the square of the largest side (2809).

Therefore, since the sum of the squares of the smaller sides is less than the square of the largest side, the triangle can be classified as an obtuse triangle.