Consider the following statements. I. In $\triangle A B C$, if $c=6$ and $\cos C=\frac{-11}{25}$, then $$…
Consider the following statements.
I. In $\triangle A B C$, if $c=6$ and $\cos C=\frac{-11}{25}$, then
$$
R=\frac{25}{2 \sqrt{14}}
$$
II. In $\triangle A B C$, if $a=3, b=4, c=6$, then $A B C$ is acute angled triangle.
Which of the above statements is/are true?
Only I
Only II
Both I and II
Neither I nor II
Solution
Given statements,
I. In $\triangle A B C$, if $c=6$ and $\cos C=-\frac{11}{25}$.
Then, $\sin C=\sqrt{1-\frac{121}{625}}=\sqrt{\frac{625-121}{625}}$
$
=\sqrt{\frac{504}{25}}=\frac{6 \sqrt{14}}{25}
$
$
\because \quad \frac{c}{\sin C}=2 R \Rightarrow R=\frac{6}{2 \times \frac{6 \sqrt{14}}{25}}=\frac{25}{2 \sqrt{14}}
$
So, statement (I) is true.
II. In $\triangle A B C$, if $a=3, b=4, c=6$, then
$
\begin{aligned}
& \cos C=\frac{a^2+b^2-c^2}{2 a b}=\frac{9+16-36}{2 \times 3 \times 4}=-\frac{11}{24} \\
\because & \cos C < 0
\end{aligned}
$
$\therefore \quad \triangle A B C$ is a obtuse angled triangle.
So, Statement (II) is false