First slide

Properties of triangles

Question

If the Cosines of the angles of a triangle are proportional to opposite sides, then the triangle is

Moderate

A

Isosceles
B

Right angled
C

Equilateral
D

Right angled isosceles

Solution

$\begin{array}{l} G i v e n \frac{C o s A}{a} = \frac{C o s B}{b} = \frac{\cos C}{c}; \\ A l s o a = 2 R \sin A, b = 2 R \sin B, c = 2 R \sin C (b y \sin e rule) \\ \Rightarrow \frac{C o s A}{2 R \sin A} = \frac{C o s B}{2 R \sin B} = \frac{\cos C}{2 R \sin C} \\ \Rightarrow \cot A = \cot B = \cot C \\ \Rightarrow A = B = C \\ \Rightarrow t h e t r i a n g l e i s e q u i l a t e r a l \end{array}$

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