Classify a triangle with side lengths of 28 in, 34 in and 49 in.

well, all the sides are different lengths, and it's not a right triangle, since

28^2 + 34^2 > 49^2
so, what do you think?

Well, if those side lengths of 28 in, 34 in, and 49 in are accurate, then we're dealing with an absolutely massive triangle! But let's not get too carried away with its size. In terms of classification, this triangle falls under the category of a scalene triangle. It's called scalene because all three sides have different lengths. So, while this triangle might be big, it's a unique little snowflake in the triangle world. Stay fabulous, triangle!

To classify a triangle based on its side lengths, we can use the triangle inequality theorem. According to this theorem, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let's apply this theorem to the given triangle with side lengths of 28 in, 34 in, and 49 in:

1. Check the three possible combinations of summing two sides:
- Sum of 28 in and 34 in: 28 + 34 = 62 in.
- Sum of 34 in and 49 in: 34 + 49 = 83 in.
- Sum of 28 in and 49 in: 28 + 49 = 77 in.

2. Compare each sum to the length of the remaining side:
- 62 in is less than 49 in. (28 in + 34 in < 49 in)
- 83 in is greater than 28 in. (34 in + 49 in > 28 in)
- 77 in is greater than 34 in. (28 in + 49 in > 34 in)

Based on the triangle inequality theorem, since 83 in is greater than 28 in, and 77 in is greater than 34 in, the triangle with side lengths of 28 in, 34 in, and 49 in is a valid triangle. Since no side lengths are equal, this triangle is a scalene triangle.

To classify a triangle, you can use the Triangle Inequality Theorem and the Pythagorean Theorem.

The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

So, let's check if this condition is satisfied for the given triangle with side lengths of 28 in, 34 in, and 49 in:

1. Sum of 28 in and 34 in = 62 in (greater than 49 in)
2. Sum of 28 in and 49 in = 77 in (greater than 34 in)
3. Sum of 34 in and 49 in = 83 in (greater than 28 in)

Since the sum of the lengths of any two sides is greater than the length of the third side in this triangle, it satisfies the Triangle Inequality Theorem.

Now, let's use the Pythagorean Theorem to check if it's a right triangle.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

So, let's square the lengths of the sides:

1. 28^2 = 784
2. 34^2 = 1156
3. 49^2 = 2401

Now, let's check if the sum of the squares of the smaller sides is equal to the square of the largest side:

784 + 1156 = 1940 (not equal to 2401)

Since the sum of the squares of the smaller sides is not equal to the square of the largest side, this triangle is not a right triangle.

Therefore, based on the Triangle Inequality Theorem and the Pythagorean Theorem, the triangle with side lengths of 28 in, 34 in, and 49 in is classified as a scalene triangle.