Question

if (-4,0) and (4,0) are two vertices of an equilateral triangle find the coordinates of the third vertex

Answer 1

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Answer 2

Answer:

The coordinates of third point are either $(0,4\sqrt{3})\text{ or }(0,-4\sqrt{3})$.

Step-by-step explanation:

Two vertices of the triangle are A(-4,0) and B(4,0).

Both points lies on the x-axis and equidistant from the origin. Since given triangle is an equilateral triangle, therefore the third vertex lie on y-axis.

Let the third vertex be C(0,y).

Using distance formula,

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$AC=\sqrt{(0-(-4))^2+(y-0)^2}=\sqrt{16+y^2}$

$AB=\sqrt{(4-(-4))^2+(0-0)^2}=\sqrt{64}=8$

Triangle ABC is an equilateral triangle and all the sides of an equilateral triangle are same.

$AC=AB$

$\sqrt{16+y^2}=8$

Squaring both sides.

$16+y^2=64$

$y^2=64-16$

Square root both the sides.

$y=\pm \sqrt{48}$

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

Therefore the coordinates of third point are either $(0,4\sqrt{3})\text{ or }(0,-4\sqrt{3})$.