Q)  If (-5,3) coordinates and (5,3) are two vertices of an equilateral triangle, then find of the third vertex, given that origin lies inside the triangle. (Take √3 = 1.7)

Ans: 

Let the third vertex be (x, y)
Hence, 3 vertex of the triangle will be A (-5,3)         B (5,3)         C (x y)

Step 1:

Since it is giiven that the triangle is equilateral, hence all the 3 sides will be equal

Hence, AC = BC = AB … (i)

AB =

AC =

BC =

Step 2:

From AC = BC in equation (i), we get:

By squaring on both sides, we get:

(-5 – x)² + (3 – y)² = (5 – x)² + (3 – y)²

(-5 – x)² = (5 – x)²

x² + 10 x + 25 = x² – 10 x + 25

20 x = 0

x = 0

Step 3:

From BC = AB in equation (i), we get:

By squaring on both sides, we get:

(5 – X)² + (3 – y)² = 100

By substituting X = 0 in this equation, we get:

25 + (3 – y)² = 100

(3 – y)² = 75

(3 – y) = ± 5√3

y = 3 – 5√3 = – 5.5 (given that √3 = 1.7)

or

y = 3 + 5√3 = 11.5 (given that √3 = 1.7)

Hence, coordinates will be (0, – 5.5) or (0, 11.5)

Point (0, 11.5) will lie on positive side of Y – axis and origin will be outside of the triangle, hence X (0, 11.5)

Point (0, – 5.5) lies on negative side of Y – axis and origin will be inside of the triangle, It satisfies the given condition,

Therefore, the coordinates of the third vertex are (0, – 5.5)