Question

if two vertices of an equilateral triangle be (0,0) (3,√3) find the third side​

Answer 1

(0,3)right solution for this question

welcome

Answer 2

Answer:

The possible coordinates of third vertex are : (0 , 2√3) or (3 , -√3)

Step-by-step explanation:

Let the two given vertices of equilateral triangle are :

A : (0 , 0)

B : (3 , √3)

Let the third vertex be C (x , y)

Now, since all the sides of an equilateral triangle are equal

⇒ BC = AC = AB

⇒ BC² = AC² = AB²

⇒ (x - 3)² + (y - √3)² = (x - 0)² + (y - 0)² = (3 - 0)² + (√3 - 0)²

⇒ (x - 3)² + (y - √3)² = x² + y² = 9 + 3

⇒ (x - 3)² + (y - √3)² = x² + y² = 12

First take, x² + y² = 12 .........(1)

And, (x - 3)² + (y - √3)² = 12 ...........(2)

⇒ x² + 9 - 6x + y² + 3 - 2√3y = 12

⇒ x² + y² - 6x + 12 - 2√3y = 12

⇒ 12 - 6x + 12 - 2√3y = 12

⇒ - 6x + 12 - 2√3y = 0

$\implies y =\frac{12-6x}{2\sqrt{3}}$

⇒ y = 2√3 - √3x.............(3)

Putting this value in equation (1)

x² + (2√3 - √3x)² = 12

⇒ 4x² - 12x = 0

⇒ x = 0 or x = 3

Now, from equation (3)

When x = 0 then y = 2√3

When x = 3 then y = -√3

Thus, the possible coordinates of third vertex are : (0 , 2√3) or (3 , -√3)