Question

Two vertices of \(\Delta ABC\) are A(-1, 4) and B(5, 2) and its centroid is G(0, \(-3\)). The coordinate of C is

• A. (4, 3)
• B. (4, 15)
• C. (-4, -15)
• D. (-15, -4)

Hint:

To quickly double-check your centroid coordinates, remember that the sum of the coordinates of all three vertices must be exactly 3 times the coordinates of the centroid:
Sum of x-coordinates: \(-1 + 5 + (-4) = 0 = 3 \times 0\)
Sum of y-coordinates: \(4 + 2 + (-15) = -9 = 3 \times (-3)\).
This serves as an instant verification method in coordinate geometry problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question provides two vertices of a triangle, \(A(-1, 4)\) and \(B(5, 2)\), along with the coordinates of its centroid, \(G(0, -3)\).
We need to determine the coordinates of the third vertex, \(C(x_3, y_3)\).

Step 2: Key Formula or Approach:
In a coordinate plane, if a triangle has vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), then its centroid \(G(x_g, y_g)\) is given by the formulas:
\[ x_g = \frac{x_1 + x_2 + x_3}{3} \]
\[ y_g = \frac{y_1 + y_2 + y_3}{3} \]

Step 3: Detailed Explanation:
$\bullet$

Step 1: Set up the coordinate values:
Given:
\(x_1 = -1\), \(y_1 = 4\) (Vertex A)
\(x_2 = 5\), \(y_2 = 2\) (Vertex B)
\(x_g = 0\), \(y_g = -3\) (Centroid G)
Let the coordinates of the third vertex \(C\) be \((x_3, y_3)\).
$\bullet$

Step 2: Solve for \(x_3\) (the x-coordinate):
Using the centroid formula for the x-coordinate:
\[ 0 = \frac{-1 + 5 + x_3}{3} \]
Multiply both sides by 3:
\[ 0 = 4 + x_3 \]
\[ x_3 = -4 \]
$\bullet$

Step 3: Solve for \(y_3\) (the y-coordinate):
Using the centroid formula for the y-coordinate:
\[ -3 = \frac{4 + 2 + y_3}{3} \]
Multiply both sides by 3:
\[ -9 = 6 + y_3 \]
\[ y_3 = -9 - 6 \]
\[ y_3 = -15 \]
Thus, the coordinates of vertex \(C\) are \((-4, -15)\).

Step 4: Final Answer:
The coordinate of vertex C is (-4, -15).