Question

lf two sides of a triangle are equal and the angle included by these sides is ${60}^0$, then the triangle is?

• A. isosceles
• B. right angled
• C. equilateral
• D. right angled isosceles

Answer 1

The correct option is C equilateral

Let $a,b,c$ be the sides of a triangle and $A,B,C$ be the angles contained between them

By $\text{sine law}$,

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

Given, $b=c$ and $A=\frac{\pi }{3}$

Therefore, $\left(2R\right)\times \left(\sin B\right)=\left(2R\right)\times \sin \left(C\right)$

$\Rightarrow \sin B=\sin C$

$\Rightarrow \text{B=C or B=}$ $\pi -C$

\sin ce, $A+B+C=\pi$ & $A=\frac{\pi }{3}$

$\Rightarrow B=C=\frac{\pi }{3}$

Now, $A=B=C$

$\Rightarrow \sin A=\sin B=\sin C$

$\Rightarrow a=b=c$

Hence, proved that $\Delta$ABC is an equilateral triangle.