iF (-4,3) and (4,3) are two vertices of an equilateral triangle ,find the coordinates of the third vertex given that - Maths - Coordinate Geometry

Question

iF (-4,3) and (4,3) are two vertices of an equilateral triangle
,find the coordinates of the third vertex given that the origin
lies in the interior of the triangle

Ajanta Trivedi · Accepted Answer

let the given vertices of an equilateral triangle be A(-4,3) and B(4,3).

since y-axis is the perpendicular bisector of AB, therefore point C will lie on y-axis

let the coordinates of the third vertex be C(0,y).

$A B^{2} = (- 4 - 4)^{2} + (3 - 3)^{2} A B^{2} = 8^{2} + 0 = 64$

since ABC is an equilateral triangle.

$A C^{2} = A B^{2} = B C^{2}$

$∵ A C^{2} = A B^{2} (0 + 4)^{2} + (y - 3)^{2} = 64 16 + y^{2} + 9 - 6 y = 64 y^{2} - 6 y - 39 = 0........ (1)$

solving the quadratic equation (1) by the formula:

$y = \frac{6 \pm \sqrt{36 + 156}}{2} = \frac{6 \pm 8 \sqrt{3}}{2} = 3 \pm 4 \sqrt{3}$

since the origin lies in the interior of the triangle therefore

$y = 3 + 4 \sqrt{3}$

hence the coordinates of the third vertex is $(0, 3 + 4 \sqrt{3})$ .

iF (-4,3) and (4,3) are two vertices of an equilateral triangle ,find the coordinates of the third vertex given that the origin lies in the interior of the triangle.