**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}

x^{2} + 10 x + 25 = x^{2} – 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)**