Question

question 4 of 5
select the correct answer from the drop - down menu.
how does extending vertex a further to the left of vertex b verify the relationship between sides and angles in triangles?
as the length ( overline{ab} ) increases, the measure of ( angle c )

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines states that \(c^{2}=a^{2}+b^{2}-2ab\cos C\), where \(c = AB\), \(a = BC\), \(b = AC\).

Step2: Analyze the effect of increasing \(AB\)

If we keep \(BC\) and \(AC\) constant (i.e., \(a\) and \(b\) are fixed) and increase \(c\) (the length of \(AB\)). Then, from the formula \(c^{2}=a^{2}+b^{2}-2ab\cos C\), we can rewrite it as \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). As \(c\) increases, \(a^{2}+b^{2}-c^{2}\) decreases. Since \(y = \cos x\) is a decreasing function for \(x\in[0,\pi]\), when \(\cos C\) decreases, \(C\) (the measure of \(\angle C\)) increases.

Answer:

Increases