Question

if (x1,y1) , (x2,y2) , (x3,y3) are the vertices of an equilateral triangle such that (x1-2)square +(y1-3)square =(y2 -3)square then x1+x2+x3+2(y1,y2,y3)=

Answer 1

Correct Question :- if (x1,y1) , (x2,y2) , (x3,y3) are the vertices of an equilateral triangle such that (x1 - 2)² + (y1 - 3)² = (x2 - 2)² + (y2 - 3)² = (x3 - 2)² + (y3 - 3)² then x1 + x2 + x3 + 2(y1 + y2+ y3) = ?

Solution :-

we know that,

• Distance between (a,b) and (c,d) = √{(a - c)² + (b - d)}
• Coordinates of centroid of ∆ = (x1 + x2 + x3)/3 and (y1 + y2 + y3)/3 .

assuming coordinates of centroid of given ∆ are G(x, y) .

since centroid is equidistant from all three vertices .

so,

→ AG = √{(x1 - x)² + (y1 - y)²}

→ AG² = {(x1 - x)² + (y1 - y)²}

similarly,

→ BG² = {(x2 - x)² + (y2 - y)²}

→ CG² = {(x3 - x)² + (y3 - y)²}

then,

→ AG² = BG² = CG²

→ {(x1 - x)² + (y1 - y)²} = {(x1 - x)² + (y1 - y)²} = {(x3 - x)² + (y3 - y)²}

given that,

→ {(x1 - 2)² + (y1 - 3)²} = {(x1 - 2)² + (y1 - 3)²} = {(x3 - 2)² + (y3 - 3)²}

comparing we get,

→ Coordinates of centroid , G = (x , y) = (2, 3)

therefore,

→ 2 = (x1 + x2 + x3)/3

→ x1 + x2 + x3 = 2 * 3 = 6

and,

→ 3 = (y1 + y2 + y3)/3

→ y1 + y2 + y3 = 3 * 3 = 9

hence,

→ x1 + x2 + x3 + 2(y1 + y2+ y3) = 6 + 2 * 9 = 6 + 18 = 24 (Ans.)

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