Question

Suppose a triangle has sides a, b, and c with side c the longest side, and that a^2 + b^2 > c^2. Let theta be the measure of the angle opposite the side of length c. Which of the following must be true? Check all that apply.

the triangle in question is a right triangle
cos(theta) <0
the triangle is not a right triangle
theta is an acute angle

Answer 1

For special triangles like right triangles, the longest side which is the hypotenuse is always the longest side. The pythagorean theorem that relates all three sides of the triangles is

a²+b²=c²

But since the problem states that the sum of the squares of a and b is greater than the square of the hypotenuse, the c is not the longest side of the triangle anymore. Thus, the triangle IS NOT a triangle. 

About the value of the angle θ, there is no way of knowing it without any given data of the sides. So, the only conclusion we can make is the nature of the triangle.