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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)
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Solution
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
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