ACUTE
To determine whether a triangle with side lengths of 48, 64, and 78 is acute, obtuse, or right, we can use the Pythagorean theorem and the properties of right triangles.
If a triangle is a right triangle, the Pythagorean theorem applies, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the length of the longest side (hypotenuse).
So, we can start by checking if this condition is met for the given triangle:
48² + 64² = 2304 + 4096 = 6400
78² = 6084
Since 6400 is greater than 6084, we can see that the given triangle does not satisfy the Pythagorean theorem, which means that it is not a right triangle.
Next, we can check whether the triangle is acute or obtuse by looking at the relationship between the square of the longest side and the sum of the squares of the other two sides. In an acute triangle, the square of the longest side is less than the sum of the squares of the other two sides, while in an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides.
So, let's compare these values for the given triangle:
48² + 64² = 6400
78² = 6084
Since 6400 is greater than 6084, we can see that the sum of the squares of the two shorter sides is less than the square of the longest side, which means that the given triangle is an ACUTE triangle.
Therefore, the triangle with side lengths of 48, 64, and 78 is an ACUTE triangle.