Final answer:
Options (a) equilateral triangle, (b) the triangle inequality theorem, and (c) an acute triangle all are valid conditions under which a triangle can be constructed. Option (d) mentioning the hypotenuse is specific to right triangles and does not guarantee a triangle can be constructed without further information.
Step-by-step explanation:
When considering the conditions for constructing a triangle, we apply the fundamental properties of triangles. Specifically:
- For a triangle with all three sides having equal lengths, we refer to it as an equilateral triangle. This is a valid type of triangle.
- The sum of any two sides being greater than the third side is a universal condition required for any triangle. If this condition is not met, a triangle cannot be formed because the sides cannot connect to enclose a space.
- Having all angles acute means that each angle is less than 90 degrees. This is characteristic of an acute triangle, which is, again, a possible configuration for a triangle.
- The statement that the longest side is the hypotenuse applies only to right triangles. The hypotenuse is opposite the right angle, and by the Pythagorean theorem, it must satisfy the equation a² + b² = c², where 'a' and 'b' are the legs, and 'c' is the hypotenuse. This condition alone does not indicate the possibility of a triangle as it does not provide enough information about the other sides or angles.
In conclusion, options (a) equilateral triangle, (b) the triangle inequality theorem, and (c) an acute triangle all describe conditions where a triangle can be constructed. Option (d) mentioning the hypotenuse, refers specifically to right triangles but does not by itself guarantee that a triangle can be constructed without additional information.