Question

The coordinates of the centroid of the triangle whose vertices are \((1, 3)\), \((-2, 7)\) and \((5, -3)\) is :

• A. \((2, \frac{7}{2})\)
• B. \((\frac{4}{3}, \frac{7}{3})\)
• C. \((4, 7)\)
• D. \((\frac{5}{3}, \frac{7}{3})\)

Hint:

The centroid of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is found by averaging the coordinates: Centroid = \( \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right) \). 1. Vertices: \((1, 3), (-2, 7), (5, -3)\). 2. x-coordinate of centroid: \(\frac{1 + (-2) + 5}{3} = \frac{1-2+5}{3} = \frac{4}{3}\). 3. y-coordinate of centroid: \(\frac{3 + 7 + (-3)}{3} = \frac{3+7-3}{3} = \frac{7}{3}\). 4. Centroid: \((\frac{4}{3}, \frac{7}{3})\).

Answer 1

The Correct Option is B

Concept: The centroid of a triangle is the point of intersection of its medians. If the vertices of the triangle are \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), then the coordinates of the centroid \((G_x, G_y)\) are given by the formulas: \[ G_x = \frac{x_1 + x_2 + x_3}{3} \] \[ G_y = \frac{y_1 + y_2 + y_3}{3} \] Step 1: Identify the coordinates of the vertices Let the vertices be: \((x_1, y_1) = (1, 3)\) \((x_2, y_2) = (-2, 7)\) \((x_3, y_3) = (5, -3)\) Step 2: Calculate the x-coordinate of the centroid (\(G_x\)) \[ G_x = \frac{1 + (-2) + 5}{3} \] \[ G_x = \frac{1 - 2 + 5}{3} \] \[ G_x = \frac{-1 + 5}{3} \] \[ G_x = \frac{4}{3} \] Step 3: Calculate the y-coordinate of the centroid (\(G_y\)) \[ G_y = \frac{3 + 7 + (-3)}{3} \] \[ G_y = \frac{3 + 7 - 3}{3} \] \[ G_y = \frac{10 - 3}{3} \] \[ G_y = \frac{7}{3} \] Step 4: Write the coordinates of the centroid The coordinates of the centroid are \((G_x, G_y) = \left(\frac{4}{3}, \frac{7}{3}\right)\). This matches option (2).