Based on the Triangle Inequality Theorem, which of the following types of triangles is possible?(1 point) Responses obtuse equilateral obtuse equilateral right isosceles right isosceles equilateral scalene equilateral scalene right equilateral
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
1. Obtuse equilateral triangle: This is not possible because an equilateral triangle has three equal sides and all angles are 60 degrees, which is less than 90 degrees. The sum of any two sides will always be greater than the third side, so it violates the Triangle Inequality Theorem.
2. Right isosceles triangle: This is possible. In a right isosceles triangle, two sides are equal in length and the third side (the hypotenuse) is longer. The sum of the two equal sides will always be greater than the length of the hypotenuse, so it satisfies the Triangle Inequality Theorem.
3. Equilateral triangle: This is possible. An equilateral triangle has three equal sides, so the sum of any two sides will always be greater than the length of the remaining side, satisfying the Triangle Inequality Theorem.
4. Scalene equilateral triangle: This is not possible because a scalene triangle has three sides of different lengths, and an equilateral triangle has three sides of equal length.
5. Right equilateral triangle: This is not possible because an equilateral triangle has three equal angles of 60 degrees, so it can't have a right angle of 90 degrees. The sum of the two equal sides will always be greater than the length of the remaining side, so it satisfies the Triangle Inequality Theorem.
Therefore, the possible types of triangles are:
- Right isosceles
- Equilateral
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Based on this theorem, the following types of triangles are possible:
1. Obtuse scalene triangle: This triangle has one angle greater than 90 degrees and no equal sides.
2. Equilateral triangle: This triangle has all three sides and angles equal.
3. Right isosceles triangle: This triangle has one right angle (90 degrees) and two equal sides.
4. Right equilateral triangle: This triangle has one right angle (90 degrees) and all three sides equal.
So, the possible triangles are obtuse scalene, equilateral, right isosceles, and right equilateral.
To determine which types of triangles are possible based on the Triangle Inequality Theorem, we need to consider the relationship between the lengths of the three sides of a triangle. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.
Let's analyze each type of triangle mentioned in the question:
1. Obtuse: An obtuse triangle has one angle greater than 90 degrees. It is possible for an obtuse triangle to be formed if the sum of the lengths of the two smaller sides is greater than the length of the longest side.
2. Equilateral: An equilateral triangle has all three sides of equal length. It is possible to have an equilateral triangle since all sides are equal in length, and the sum of any two sides will always be greater than the length of the third side.
3. Right: A right triangle has one angle exactly 90 degrees. It is possible to have a right triangle if it satisfies the Pythagorean Theorem (where the square of the longest side is equal to the sum of the squares of the other two sides).
4. Isosceles: An isosceles triangle has two sides of equal length. It is possible to have an isosceles triangle as long as the sum of the lengths of the two equal sides is greater than the length of the third side.
5. Scalene: A scalene triangle has all three sides of different lengths. It is possible to have a scalene triangle if the sum of the lengths of the two smaller sides is greater than the length of the longest side.
Based on the explanations provided, the possible types of triangles are:
- Obtuse
- Equilateral
- Right
- Isosceles
- Scalene
Therefore, all the triangle types mentioned in the question (obtuse, equilateral, right, isosceles, scalene) are possible according to the Triangle Inequality Theorem.