Question

One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.

Options

• (–1, –1)

• (2, 2)

• (–2, –2)

• (2, –2)

Answer 1

One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is (–2, –2).

Explanation:

Let ABC be an equilateral triangle with vertex (x₁, y₁).

AD ⊥ BC and let (a, b) be the coordinates of D.

Given that the centroid G lies at the origin i.e., (0, 0)

Since, the centroid of a triangle,divides the median in the ratio 1 : 2

So, 0 = `(1 xx x_1 + 2 xx a)/(1 + 2)`

⇒ x₁ + 2a = 0  ......(i)

And 0 = `(1 xx y_1 + 2 xx b)/(1 + 2)`

⇒ y₁ + 2b = 0  ......(ii)

Equations of BC is given by x + y – 2 = 0  .....(iii)

Point D(a, b) lies on the line x + y – 2 = 0 

So a + b – 2 = 0

Slope of equation (iii) is = – 1

And the slope of AG = `(y_1 - 0)/(x_1 - 0) = y_1/x_1`

Since, they are perpendicular to each other

∴ `- 1 xx y_1/x_1` = – 1

⇒ y₁ = x₁

From eq. (i) and (ii) we get

x₁ + 2a = 0 

⇒ 2a = – x₁

y₁ + 2b = 0 

⇒ 2b = – y₁

∴ a = b

From equation (iv) we get

a + b – 2 = 0

⇒ a + a – 2 = 0

⇒ 2a – 2 = 0

⇒ a = 1 and b = 1   ....[∵ a = b]

∴ x₁ = – 2 × 1 = – 2

And y₁ = – 2 × 1 = – 2