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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
In a triangle ABC , if $b + c =2 a$ and $\angle A =60^{\circ}$, then $\triangle ABC$ is
  • A
    Scalene
  • ✓
    Equilateral
  • C
    Isosceles
  • D
    Right angled
✓

Answer

Correct option: B.
Equilateral
(B) We have, $b+c=2 a$ ...(i)
$\cos 60^{\circ}=\frac{b^2+c^2-a^2}{2 b c}=\frac{(b+c)^2-2 b c-a^2}{2 b c}$
$\Rightarrow \frac{1}{2}=\frac{4 a ^2-2 bc - a ^2}{2 bc } \Rightarrow \frac{1}{2}=\frac{3 a ^2}{2 bc }-1$
$\Rightarrow \frac{3}{2}=\frac{3 a ^2}{2 bc }$
$\Rightarrow bc = a ^2$ ...(ii)
From (i) and (ii), we get
$b + c =2 \sqrt{b} \sqrt{ c }$
$\Rightarrow(\sqrt{ b }-\sqrt{ c })^2=0 \Rightarrow b= c$
From (i), $a=b=c$
$\therefore \quad \triangle ABC$ is equilateral.

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