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Let (-4,3) and (4,3) are two vertices of an equilateral triangle and the origin lies in the interior of the triangle. Find out the coordinates of the third vertex.

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0, 3 + 43

0, 3 − 53

0, 3 − 43

0, 3 + 53

answer is C.

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Detailed Solution

Given vertices of equilateral triangles are

A (- 4, 3) and B (4, 3)

.
Let the coordinates of the third vertex be C

(x, y) .

We know that,
The distance between any two points in a Cartesian plane

M a, b, N c, d

is given by

M N = (c − a) 2 + (d − b) 2

.
Therefore, distance between

A (- 4, 3) and B (4, 3)

AB = (− 4 − 4) 2 + (3 − 3) 2 = (− 8) 2 = 8 units

Since, given triangle is equilateral and in equilateral triangle sides are equal.
So, the length of the side of an equilateral triangle is 8 units.
Therefore,

B C = 8 ⇒ (4 − x) 2 + (3 − y) 2 = 8 ... (1)

And

C A = 8 ⇒ (− 4 − x) 2 + (3 − y) 2 = 8 ... (2)

From equation (1) and (2),

(4 − x) 2 + (3 − y) 2 = (− 4 − x) 2 + (3 − y) 2 ⇒ (4 − x) 2 + (3 − y) 2 = (4 + x) 2 + (3 − y) 2 ⇒ 16 + x 2 − 8 x + 9 + y 2 − 6 y = 16 + x 2 + 8 x + 9 + y 2 − 6 y ⇒ − 8 x = 8 x ⇒ x = 0

On substituting

x = 0

in equation (1) we get,

(4 − 0) 2 + (3 − y) 2 = 8 ⇒ 16 + (3 − y) 2 = 64 ⇒ y 2 − 6 y − 39 = 0 ⇒ y = 6 ± 36 + 4 × 39 2 ⇒ y = 6 ± 192 2 ⇒ y = 3 ± 48 ⇒ y = 3 ± 43

It is given that the origin lies inside the triangle, which means the y-coordinate of the third vertex must be negative.

∴ y = 3 − 43

Therefore, the coordinates of the third vertex are

0, 3 − 43

.
Hence, option (3) is correct.

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