In a triangle ABC, if a A = b B, where a ≠ b, then ABC is

Correct Answer is : Right angled triangle

Correct Answer is : Right angled triangle

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AP EAPCET 06-Jul-2022 Shift 2 Paper

Section: Mathematics

Question : 28 of 160

Marks: +1, -0

In a triangle

ABC

a \cos A = b \cos B

a \neq b

\triangle ABC

Solution: 👈: Video Solution

a \cos A = b \cos B

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

\; \Rightarrow \; \frac{a}{b} (b^2+c^2-a^2) = \; \frac{b}{a} (a^2+c^2-b^2)

\; \Rightarrow a^2 (b^2+c^2-a^2) = b^2 (a^2+c^2-b^2)

\; \Rightarrow a^2 b^2 + a^2 c^2 - a^4 = a^2 b^2 + b^2 c^2 - b^4

\; \Rightarrow a^4 - b^4 + b^2 c^2 - a^2 c^2 = 0

\; \Rightarrow (a^2 - b^2)(a^2 + b^2) + c^2 (b^2 - a^2) = 0

\; \Rightarrow (a^2 - b^2)[a^2 + b^2 - c^2] = 0

\; \; \text{ As } \; (a \neq b), a^2 - b^2 \neq 0

\; a^2 + b^2 - c^2 = 0 \Rightarrow a^2 + b^2 = c^2

\therefore \triangle A B C

is a right angled triangle with

A B

as hypotenuse.

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