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

A the triangle is not a right triangle
B theta is an acute angle
C The triangle in question is a right triangle
D costheta<0

Answer 1

A the triangle is not a right triangle

D costheta <0

The law of cosines says:

a²=b²+c²−2bccos(θ)

Rewriting:

2bccos(θ)=b²+c²−a²<0

So cos(θ)<0cos(θ)<0 since 2bc>02bc>0. Since 0<θ<π0<θ<π in any triangle,

π/2<θ<π

So:

1. θ is not an acute angle.

2 The triangle is not a right triangle. In a right triangle, one of the angles is 90 degrees and the other two are then less than 90 degrees. This triangle has an angle greater than 90 degrees.

3. cos(θ)<0 is true.

4. cos(θ)>0 is false -3 says cos(θ) is negative; a number can't be both positive and negative.