Suppose a triangle has sides [tex]$a, b$[/tex], and [tex]$c$[/tex], and that [tex]$a^2+b^2\ \textless \ c^2$[/tex]. Let [tex]$\theta$[/tex] be the measure of the angle opposite the side of length [tex]$c$[/tex]. Which of the following must be true? Check all that apply.
A. [tex]$\cos \theta\ \textgreater \ 0$[/tex]
B. [tex]$a^2+b^2-c^2=2 a b \cos \theta$[/tex]
C. [tex]$\cos \theta\ \textless \ 0$[/tex]
D. The triangle is not a right triangle.

To determine which statements must be true, let's analyze the given condition and apply trigonometry:

Given Condition:


The inequality a 2 + b 2 < c 2 indicates that the triangle is not a right triangle. Specifically, this is a property of an obtuse triangle, where the side opposite the largest angle, c , is longer than any other side.


Understanding θ :


θ is the angle opposite the side c . Because a 2 + b 2 < c 2 , θ must be greater than 90 degrees. This means the triangle is obtuse.


Cosine Rule:


According to the cosine rule: a 2 + b 2 = c 2 − 2 ab cos θ .

Rearranging gives a 2 + b 2 − c 2 = − 2 ab cos θ .

Since a 2 + b 2 < c 2 , it follows that a 2 + b 2 − c 2 < 0 , which implies cos θ < 0 .



Evaluating the Options:


Option A: 0"> cos θ > 0 is false, because we've established that cos θ < 0 for an obtuse angle.

Option B: a 2 + b 2 − c 2 = 2 ab cos θ is false, as the correct expression is a 2 + b 2 − c 2 = − 2 ab cos θ .

Option C: cos θ < 0 is true, since θ is obtuse.

Option D: The triangle is not a right triangle is true, because a 2 + b 2 < c 2 indicates an obtuse triangle.


Therefore, the correct statements are:

C. cos θ < 0 implies the angle is obtuse.

D. The triangle is not a right triangle.


By following these steps, we can confidently determine the properties of the triangle based on the given condition.

Answered by danjohnbrain | 2025-07-07