In a A B C, if a²+b²+c²=a c+{3} a b, then the triangle is

Correct Answer is : right angled and not isosceles

Correct Answer is : right angled and not isosceles

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Manipal Entrance Test 2015 Math Paper

Question : 48 of 70

Marks: +1, -0

In a

\triangle A B C,

a^{2}+b^{2}+c^{2}=a c+\sqrt{3} a b,

then the triangle is

Solution:

We have,

a^{2}+b^{2}+c^{2}=a c+\sqrt{3} a b

⇒

\frac{a^{2}}{4}+c^{2}-a c+\frac{3 a^{2}}{4}

+b^{2}-\sqrt{3} a b=0

⇒

\left(\frac{a}{2}-c\right)^{2}+\left(\frac{\sqrt{3} a}{2}-b\right)^{2}=0

⇒

\frac{a}{2}-c=0

and

\frac{\sqrt{3}}{2} a-b=0

⇒

a=2 c

and

\sqrt{3} a=2 b

⇒

a=\frac{2 b}{\sqrt{3}}=2 c=\lambda

[say]

⇒

a=\lambda, b=\frac{\sqrt{3}}{2} \lambda

and

c=\frac{\lambda}{2}

Now,

b^{2}+c^{2}= \left(\frac{\sqrt{3} \lambda}{2}\right)^{2}+ \left(\frac{\lambda}{2}\right)^{2}

=\frac{3 \lambda^{2}}{a}+\frac{\lambda^{2}}{4}=\lambda^{2}

∴

b^{2}+c^{2}=a^{2}

Hence, the triangle is right angled but not isosceles.

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