Based on the Triangle Inequality Theorem, which of the following types of triangles is possible?(1 point)
Responses
equilateral scalene
right isosceles
right equilateral
obtuse 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 only possible type of triangle from the given options is a scalene triangle. This is because an equilateral triangle has all sides equal, and for a right isosceles triangle, the two equal sides would have to be longer than the hypotenuse, which violates the triangle inequality. Similarly, an obtuse equilateral triangle would have all sides equal, but the sum of the lengths of any two sides would not be greater than the length of the third side.
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:
- An equilateral triangle can satisfy the inequality because all sides are equal in length.
- A scalene triangle can also satisfy the inequality because no sides are equal in length.
- A right isosceles triangle can satisfy the inequality because the two equal sides (legs) would be longer than the length of the hypotenuse.
- However, a right equilateral triangle cannot exist because it would violate the Triangle Inequality Theorem. In an equilateral triangle, all sides are equal in length, but in a right triangle, the hypotenuse is always longer than the legs.
- An obtuse equilateral triangle cannot exist because the Triangle Inequality Theorem is violated. In an obtuse triangle, the longest side is the one opposite the obtuse angle. In an equilateral triangle, all sides are equal in length, so it is not possible to have a triangle where one side is longer than the other two.
Therefore, the possible types of triangles based on the Triangle Inequality Theorem are equilateral, scalene, and right isosceles.