Question

If the centroid and a vertex of an equilateral triangle are 2,3 and 4,3

Answer 1

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Answer 2

The other coordinates of the triangle are (1, 3+√3) and (1, 3-√3)

Given,

Let the triangle under consideration be ΔABC

The centroid of the triangle, G = (2,3)

One vertex of the triangle, A = (4,3)

To Find,

The other two vertices of the triangle (B=? and C=?)

Solution,

Consider the triangle ΔABC

Let AD be the median of the triangle.

We know that the centroid G divides the median of a triangle in the ratio of 2:1

⇒ AG/GD = 2/1

⇒ AG = 2GD

Now let the coordinates of D be (x,y)

Then, G = ((2x+4)/3, (2y+3)/3)

⇒ (2,3) = ((2x+4)/3, (2y+3)/3)

⇒ 2 = (2x+3)/3 and 3 = (2y+3)/3

⇒ 6 = 2x+3 and 9 = 2y+3

⇒ 2x = 6-3 and 2y = 9-3

⇒ x = 3/2 and y = 3

Therefore, the vertex D is (3/2, 3)

Now, As ΔABC is an equilateral triangle, so median is the same as the altitude of the triangle.

⇒ tan 60° = AD/BD (each angle in an equilateral triangle is 60°)

⇒ √3 = 3/BD

⇒ BD = 3/√3

⇒ BD = √3

Now, A(4,3), G(2,3), and D(1,3) lie on y = 3, and BC is perpendicular to AD.

⇒ B, C, and D have the same x-coordinate = 1

Then √((1−1)² + (y−3)²)  = √3

⇒ (y-3)² = 3

⇒ y = 3 ± √3

Therefore, the other coordinates of the triangle are B(1, 3+√3) and C(1, 3-√3)

Correction in question: If the centroid and a vertex of an equilateral triangle are (2,3) and (4,3), then find the other coordinates of the triangle.

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