If x2-y2-z2-x y-y z-z x- then the triangle is -A- IsoscelesB- ScaleneC- Right angledD- Equilateral

The correct option is D Equilateralx^2 + y^2 + z^2 = xy + yz + zx ⇒ x^2 + y^2 + z^2 xy yz zx = 0 ⇒ 2(x^2 + y^2 + z^2 xy yz zx) = 0 ⇒ (x^2 + y^2) + ...

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Quantitative Aptitude

Triangles

If x2+y2+z2=x...

Question

If $x^{2} + y^{2} + z^{2} = x y + y z + z x,$ then the triangle is :

Isosceles

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Right angled

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Equilateral

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Scalene

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Solution

The correct option is C Equilateral
$x^{2} + y^{2} + z^{2} = x y + y z + z x \Rightarrow x^{2} + y^{2} + z^{2} - x y - y z - z x = 0 \Rightarrow 2 (x^{2} + y^{2} + z^{2} - x y - y z - z x) = 0 \Rightarrow (x^{2} + y^{2}) + (y^{2} + z^{2}) + (z^{2} + x^{2}) - 2 x y - 2 y z - 2 z x = 0 \Rightarrow (x - y)^{2} + (y - z)^{2} + (z - x)^{2} = 0 \Rightarrow x = y = z$
$∴$ The given triangle is an equilateral triangle

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If x2+y2+z2=xy+yz+zx, then the triangle is :

The correct option is C Equilateralx2+y2+z2=xy+yz+zx⇒x2+y2+z2−xy−yz−zx=0⇒2(x2+y2+z2−xy−yz−zx)=0⇒(x2+y2)+(y2+z2)+(z2+x2)−2xy−2yz−2zx=0⇒(x−y)2+(y−z)2+(z−x)2=0⇒x=y=z ∴ The given triangle is an equilateral triangle

If $x^{2} + y^{2} + z^{2} = x y + y z + z x,$ then the triangle is :