The orthocenter of an equilateral triangle is | vedclass

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(A) Let the orthocenter be $O = (3, -2)$.Since one side lies on the $x$-axis,the altitude from the vertex $A$ to this side is perpendicular to the $x$

Vedclass · Accepted Answer

$(3, -6)$

Vedclass · Answer

(A) Let the orthocenter be $O = (3, -2)$.Since one side lies on the $x$-axis,the altitude from the vertex $A$ to this side is perpendicular to the $x$-axis.The foot of the altitude $M$ on the $x$-axis is $(3, 0)$.The distance $OM$ is the vertical distance between $(3, -2)$ and $(3, 0)$,so $OM = |-2 - 0| = 2$.In an equilateral triangle,the orthocenter divides the altitude in the ratio $2:1$.Thus,$\frac{AO}{OM} = \frac{2}{1}$,which implies $AO = 2 	imes OM = 2 	imes 2 = 4$.The total length of the altitude $AM = AO + OM = 4 + 2 = 6$.Since the vertex $A$ is at a distance of $6$ units from the $x$-axis and lies on the line $x = 3$,the coordinates of $A$ are $(3, -6)$.

The orthocenter of an equilateral triangle is $(3, -2)$. If one of its sides lies on the $x$-axis,find the vertex of the triangle that does not lie on the $x$-axis.

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