Question

the vertex of an equilateral triangle is at 2, - 1 and the opposite side to it has the equation X + Y is equals to 2 then the orthocentre of the triangle is​

Answer 1

Answer:

Step-by-step explanation:In an equilateral triangle, the centroid, orthocenter, etc. coincide at the same point. So if we find the centroid, we will be done.

The equation of the perpendicular to the third side will be y=x and cuts it at (0.5,0.5). This line will also be a median of the triangle. The centroid divides the median in the ratio 2:1 from the vertex. Using section formula,

G(13,13)

The formula for the coordinates of the centroid of a triangle with vertices Ai(xi,yi)i=1,2,3 is

G(∑xi3,∑yi3)

But notice that one of the vertices is the origin. So we only have to find the sum of the abscissae and ordinates of the other two vertices to get the answer. So let's proceed analytically.

The perpendicular from the origin to the line will be one of the heights of the triangle. The length of perpendicular is given by the following formula,

p=|0+0−1|2–√

Now using basic trigonometry, the side of the triangle will be equal to 26√.

The circle centred at origin having this radius cuts the line x+y=1 at two distinct points which will be the other two vertices of our triangle.

x2+y2=23

x2+(1−x)2=23

2x2−2x+13=0

Sum of roots =1=x2+x3

From the equation of the line we have,

x2+y2=1 and x3+y3=1

or y2+y3=1

Just put it in the centroid formula,

G(13,13)

Use the parametric form of a line. Then the two vertices will be

(±16√⋅12√+12,±16√⋅12√+12)

Subsequently, the answer.

G(13,13)