1. A triangle has side lengths of 8, 10, and 12. what kind of triangle is it?

2. A triangle has side lengths of 3, 4, and 7. what kind of triangke is it?

take the two smaller sides, call them a and b. call the longest side c.

if a^2 + b2 = c^2 you have a right triangle

if < c^2 it's acute
if > c^2 it's obtuse

in addition, if you can find a,b,c such that

a=b=c it's equilateral
a=b≠c it's isosceles
a≠b≠c it's scalene

plug in your numbers and make your choice.

I just wanted an answer not an explanation

1. Well, that triangle is no joker! With side lengths of 8, 10, and 12, it belongs to the classic "Scalene Triangle" gang. Unlike isosceles or equilateral triangles, this bad boy has no equal sides. Triangles really like to keep us on our toes!

2. Ah, this triangle must have left the sideshop without the rulebook! With side lengths of 3, 4, and 7, it doesn't fit into any of the typical categories. In fact, it's a special type called an "Obtuse Scalene Triangle." The word "obtuse" means it has one angle measuring more than 90 degrees, while "scalene" means all three sides have different lengths. Talk about a unique character!

1. To determine the type of triangle, we can check the relationships between its side lengths. In this case, we have a triangle with side lengths 8, 10, and 12.

To find out if it is a right triangle, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Let's check if 8^2 + 10^2 = 12^2:
64 + 100 = 144

Since 64 + 100 is not equal to 144, we can conclude that the triangle is not a right triangle.

Next, we can determine if it is an equilateral triangle. In an equilateral triangle, all three sides are equal.

Since the given triangle has sides of different lengths (8, 10, and 12), we can conclude that it is not an equilateral triangle.

Finally, let's check if it is an isosceles triangle. In an isosceles triangle, at least two sides are equal.

Since the given triangle does not have any two sides with the same length, we can conclude that it is not an isosceles triangle either.

Therefore, the triangle with side lengths 8, 10, and 12 is classified as a scalene triangle, which means that all three sides have different lengths.

2. Now, let's analyze the triangle with side lengths 3, 4, and 7.

First, let's use the same process to check if it is a right triangle. We can use the Pythagorean theorem:

3^2 + 4^2 = 9 + 16 = 25

Since 25 is equal to 7^2, we can conclude that the triangle is a right triangle.

Next, let's check if it is an equilateral triangle. In an equilateral triangle, all three sides are equal. However, since the sides of the given triangle have different lengths (3, 4, and 7), we can conclude that it is not an equilateral triangle.

Finally, let's check if it is an isosceles triangle. In an isosceles triangle, at least two sides are equal. Since the given triangle does not have any two sides with the same length, it is not an isosceles triangle either.

To summarize, the triangle with side lengths 3, 4, and 7 is classified as a right scalene triangle. It is a right triangle because it satisfies the Pythagorean theorem, and it is scalene because all three sides have different lengths.

To determine the type of triangle based on its side lengths, we need to consider the relationships between the lengths of the sides. Here are the steps to determine the type of triangle:

1. Examine the lengths of the three sides.
2. Determine if any of the sides are equal to each other.

Let's apply these steps to the given triangles:

1. Triangle with side lengths 8, 10, and 12:
- Comparing the lengths, we don't find any sides that are equal.
- Since none of the sides are equal, we know that it is not an equilateral triangle.
- Next, check if any two sides added together are greater than the third side.
- In this case, 8 + 10 is greater than 12, 10 + 12 is greater than 8, and 12 + 8 is greater than 10.
- Since this condition is true for all three pairs of sides, it is a valid triangle.
- Finally, since none of the sides are equal, it is not an isosceles triangle.
- Based on the given information, this triangle is a scalene triangle.

2. Triangle with side lengths 3, 4, and 7:
- Comparing the lengths, we observe that none of the sides are equal.
- Since none of the sides are equal, we already establish that it is not an equilateral or isosceles triangle.
- Next, let's check if any two sides added together are greater than the third side.
- In this case, 3 + 4 is greater than 7, 4 + 7 is also greater than 3, but 7 + 3 is not greater than 4.
- Since this condition is not true for the pair of sides 7 and 3, it violates the triangle inequality theorem.
- Therefore, triangle inequality is not satisfied, and this triangle cannot exist.

In summary:
1. The triangle with side lengths 8, 10, and 12 is a scalene triangle.
2. The triangle with side lengths 3, 4, and 7 does not form a valid triangle since it violates the triangle inequality theorem.