Question

Consider the following statements.
I. In $\Delta A B C$, 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?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

Answer: (A)

Given statements,
I. In $\Delta A B C$, if $c=6$ and $\cos C=-\frac{11}{25}$.
Then, $\sin\, C =\sqrt{1-\frac{121}{625}}=\sqrt{\frac{625-121}{625}}$
$=\sqrt{\frac{504}{25}}=\frac{6 \sqrt{14}}{25}$
$\because\frac{c}{\sin C}=2 R$
$\Rightarrow R=\frac{6}{2 \times \frac{6 \sqrt{14}}{25}}$
$=\frac{25}{2 \sqrt{14}}$
So, statement (I) is true.
II. In $\triangle A B C$, if $a=3, b=4, c=6$, then
$\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}=\frac{9+16-36}{2 \times 3 \times 4}$
$=-\frac{11}{24}$
$\because \cos C<0$
$\therefore \Delta A B C'$ is a obtuse angled triangle.
So, Statement (II) is false.