Question

If the centroid of the triangle ABC with vertices A(5,-4), B(7,8) and C(a,b) is G(7,6), then (b-a) is

• A. 2
• B. 5
• C. 9
• D. 14

Hint:

To quickly find a missing vertex when the centroid and two other vertices are known, use the linear shortcut: \[ \text{Missing Coordinate} = 3 \times (\text{Centroid Coordinate}) - (\text{Sum of other two coordinates}) \]

• a = 3(7) - (5 + 7) = 21 - 12 = 9

• b = 3(6) - (-4 + 8) = 18 - 4 = 14
This direct formula eliminates fractions immediately!

Answer 1

The Correct Option is B

Concept: The coordinates of the centroid G(X, Y) of a triangle with vertices (x_1, y_1), (x_2, y_2), and (x_3, y_3) are given by the average of the respective coordinates of its vertices: \[ X = \frac{x_1 + x_2 + x_3}{3}, \quad Y = \frac{y_1 + y_2 + y_3}{3} \]

Step 1: Set up the coordinate equations using the given coordinates. We are given:

• Vertices: A(5, -4), B(7, 8), and C(a, b)

• Centroid: G(7, 6)

Step 2: Solve for the unknown x-coordinate, a. Using the centroid formula for the X-coordinate: \[ 7 = \frac{5 + 7 + a}{3} \] Multiply both sides by 3: \[ 21 = 12 + a \] Isolate a: \[ a = 21 - 12 = 9 \]

Step 3: Solve for the unknown y-coordinate, b. Using the centroid formula for the Y-coordinate: \[ 6 = \frac{-4 + 8 + b}{3} \] Multiply both sides by 3: \[ 18 = 4 + b \] Isolate b: \[ b = 18 - 4 = 14 \]

Step 4: Calculate the required value of (b - a). Using the values a = 9 and b = 14: \[ b - a = 14 - 9 = 5 \] This precisely matches Option (B).