Question

If (-4,3) and (4,3) are the two vertices of an equilateral triangle find the coordinates of the third vertex given that the origin lies in the (1) interior (2) exterior of the triangle

Answer 1

On calculating the distance between the two points, we find that it is equal to 8. Also the mid point is (0,3). The distance of the third vertex from the base is √3/2 times the side. So third vertex is at a distance of

√3/2 * 8 = 4√3

from (0,3) i.e. it may be (0,3+4√3) or (0,3-4√3).

If the vertex is (0,3-4√3) then the origin lies in interior and when it is (0,3+4√3) then the origin lies in exterior.

Answer 2

Solution :---

let the Third vertices be (x,y)

then Distance between (x,y) & (4,3) is :--

→ √(x-4)² + (y-3)² ---------------- Equation (1)

and Distance between (x,y) & (-4,3) is :-----

→ √(x+4)² + (y-3)² ---------------- Equation (2)

Distance between (4,3) &(-4,3) is :-------

→ √(4+4)² + (3-3)² = 8 units. ---------------- Equation (3)

Now, since, Distance Between them all is Equal , as it is Equaliteral ∆.

so, Equation (1) = Equation (2)

→ √(x-4)² + (y-3)² = √(x+4)² + (y-3)²

→ (x-4)² = (x+4)²

→ x² - 8x + 16 = x² +8x +16

→ 16x = 0

→ x = 0

And, also , Equation (1) = Equation (3)

→ √(x-4)² + (y-3)² = 8

Squaring both sides

→ (x-4)² + (y-3)² = 64

Putting value of x = 0, now,

→ (y-3)² = 64-16

→ (y-3)² = 48

Square - root both sides now,

→ (y-3) = ±4√3

→ y = ±4√3 + 3

Now, as origin lies in the interior of the triangle,

y ≠ 3+4√3 .

∴ Third vertex = (x, y) = (0, 3 - 4√3).