Question

Show that the points (p. p), (-P. -p), (p√3,- p√3) are the vertices of an
equilateral triangle.​

Answer 1

⇝G I V E N:-

↣Three coordinates of a triangles are (p,p) (-p,-p) and (p√3,-p√3)

⇝T O P R O V E:-

↣The coordinates of vertices of triangle are of equilateral triangle

⇝S O L U T I O N:-

Let A=(p,p) ,B=(-p,-p) and C=(p√3,-p√3)

Now , using distance formula we will find the distance between :-

↣A to B

⇢AB=√(-p-p)²+(-p-p)²

⇢AB=√(-2p)²+(-2p²)

⇢AB=√4p²+4p²

⇢AB=√8p²

↣B to C

⇢BC=√(p√3+p)²+(-p√3+p)²

⇢BC=√(3p²+p²+2×p√3×p)+(3p²+p²-2×p√3×p)

⇢BC=√4p²+2p²√3+4p²-2p²√3

⇢BC=√4p²+4p²

⇢BC=√8p²

↣A to C

⇢AC=√(p√3-p)²+(-p√3-p)²

⇢AC=√(3p²+p²-2×p√3×p)+(3p²+p²+2×p√3×p)

⇢AC=√4p²-2p²√3+4p²+2p²√3

⇢AC=√4p²+4p²

⇢AC=√8p²

Here we can see that AB=BC=AC ,since all sides are equal.

∴ These given vertices are of an equilateral triangle.

Hence,proved