Question

If (-4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the (i) interior (ii) exterior of the triangle.

Answer 1

Let B (-4, 3) and C (4, 3) be the given two vertices of the equilateral triangle.

Let A (x, y) be the third vertex.

Then, we have

AB = BC = AC

Let us consider the part AB = BC

AB² = BC²

(-4 – x)² + (3 – y)² = (4 + 4)² + (3 – 3)²

16 + x² + 8x + 9 + y² – 6y = 64

x² + y² + 8x – 6y = 39

Now, let us consider AB = AC

AB² = AC²

(-4 – x)² + (3 – y)² = (4 – x)² + (3 – y)²

16 + x² + 8x + 9 + y² – 18y = 16 + x² – 8x + 9 + y² – 6y

16x = 0

x = 0

Now, BC = AC

BC² = AC²

(4 + 4)² + (3 – 3)² = (4 – 0)² + (3 – y)²

64 + 0 = 16 + 9 + y² – 6y

64 = 16 + (3 – y)²

(3 – y)² = 48

3 – y = ± 4√3

y = 3 ± 4√3

Therefore, the coordinates of the third vertex

(i) When origin lies in the interior of the triangle is (0, 3 – 4√3)

(ii) When origin lies in the exterior of the triangle is (0, 3 + 4√3)