Question

based on the triangle inequality theorem, which of the following types of triangles is possible? (1 point) equilateral scalene obtuse equilateral right equilateral right isosceles

Explanation:

Step1: Recall triangle - type definitions

An equilateral triangle has all sides equal. A scalene triangle has all sides different. So, an equilateral scalene is a contradiction.

Step2: Analyze angle - side relationships

An equilateral triangle has all angles equal to 60° (since the sum of angles in a triangle is 180° and 180°÷3 = 60°). So, it cannot be obtuse (an obtuse - angled triangle has one angle greater than 90°) or right - angled (a right - angled triangle has one angle equal to 90°).

Step3: Consider right isosceles

A right isosceles triangle has one angle equal to 90° and two equal sides. Let the equal sides be \(a\) and the hypotenuse be \(c\). By the Pythagorean theorem \(c^{2}=a^{2}+a^{2}=2a^{2}\), and it satisfies the triangle - inequality theorem (for any triangle with side lengths \(x\), \(y\), \(z\), \(x + y>z\), \(x+z > y\), and \(y + z>x\)).

Answer:

right isosceles