In a triangle ABC, a = 3, b = 5, c = 7. Find the angle opposite to C. Options: A . &nbsp 60 ∘ B . &nbsp 90 ∘ C . &nbsp 120 ∘ D . &nbsp 150 ∘

In a triangle ABC, a = 3, b = 5, c = 7. Find the angle opposite to C.

Discussion Forum : Solution Of Triangles

Question -

In a triangle ABC, a = 3, b = 5, c = 7. Find the angle opposite to C.

Options:

A .

60^{\circ}

B .

90^{\circ}

C .

120^{\circ}

D .

150^{\circ}

Answer: Option C
:
C

We know from trigonometric ratios of half angles that $t a n \frac{A}{2} = \sqrt{\frac{(s - b) (s - c)}{(s - a) (s)}}$
Here we need angle opposite to c i.e., C.
Semiperimeter, $S = \frac{a + b + c}{2} = \frac{3 + 5 + 7}{2} = \frac{15}{2}$
$t a n \frac{C}{2} = \sqrt{\frac{(s - b) (s - a)}{(s - c) (s)}}$
$= \sqrt{\frac{(s - 3) (s - 5)}{(s - 7) s}} = \sqrt{\frac{(15 - 6) (15 - 10) (2)}{4 (15 - 14) (\frac{15}{2})}}$
$= \sqrt{\frac{9 (5)}{15}} = \sqrt{3} \Rightarrow \frac{C}{2} = 60^{\circ} \Rightarrow C = 120^{\circ}$

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