Question

If two vertices of an equilateral triangle is(0,0) and(3,✓3). Find the third vertex.

Answer 1

Answer: The answer is (0, 2√3) or (3, -√3).

Step-by-step explanation: Given in the question and shown in the attached figure, ABC is an equilateral triangle. The two vertices are A(0, 0) and B(3, √3).  We are to find the co-ordinates of the third vertex 'C'.

Let the co-ordinates of the third vertex be C(a,b).

Since ΔABC is equilateral, so AB = BC = CA.

Now,

$a^2+b^2=12,$

and

$(a-3)^2+(b-\sqrt 3)^2=12,\\\\\Rightarrow a^2-6a+9+b^2-2\sqrt 3b+3=12\\\\\Rightarrow 24-6a-2\sqrt 3b=12\\\\\Rightarrow -6a-2\sqrt 3b=-12\\\\\Rightarrow a=\dfrac{6-\sqrt 3b}{3}.$

Putting this in the previous equation, we have

$(\dfrac{6-\sqrt 3b}{3})^2+b^2=12\\\\\\\Rightarrow 12b^2-12\sqrt 3b-72=0\\\\\Rightarrow (b-2\sqrt 3)(b+\sqrt 3)=0\\\\\Rightarrow b=2\sqrt 3,~~b=-\sqrt 3.$

If b = 2√3, then a = 0

If b = - √3, a = 3

Thus, the third vertex of the equilateral triangle is (0, 2√3) or (3, -√3).

Answer 2

Answer:

Step-by-step explanation: