Question

If $G(4,3,3)$ is the centroid of the triangle $ABC$ whose vertices are $A(a,3,1)$, $B(4,5,b)$ and $C(6,c,5)$, then the values of $a$, $b$, $c$ are

• A. $a = 1, b = 2, c = 3$
• B. $a = 3, b = 2, c = 1$
• C. $a = 2, b = 1, c = 3$
• D. $a = 2, b = 3, c = 1$

Hint:

Isolate and test a single variable from the given options to filter out answers instantly. Solving for the first coordinate gives $a = 2$, which immediately eliminates options (A) and (B) without requiring any additional algebra.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the 3D coordinate values of the centroid $G(4,3,3)$ of a triangle $ABC$. The definitions of its vertices contain individual unknown parameters: $A(a,3,1)$, $B(4,5,b)$, and $C(6,c,5)$. Our goal is to extract the exact values of $a$, $b$, and $c$.

Step 2: Key Formula or Approach:
The coordinates of the centroid $G(x,y,z)$ of a triangle with vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, and $(x_3, y_3, z_3)$ are given by the average of the respective component positions: $$x = \frac{x_1 + x_2 + x_3}{3}, \quad y = \frac{y_1 + y_2 + y_3}{3}, \quad z = \frac{z_1 + z_2 + z_3}{3}$$ We can map each component independently to find the values of $a$, $b$, and $c$.

Step 3: Detailed Explanation:
Let's substitute the coordinates of $A$, $B$, $C$, and $G$ into the centroid component formulas: 1. Equating the $x$-coordinates: $$4 = \frac{a + 4 + 6}{3}$$ Multiply by 3: $$12 = a + 10 \implies a = 2$$ 2. Equating the $y$-coordinates: $$3 = \frac{3 + 5 + c}{3}$$ Multiply by 3: $$9 = 8 + c \implies c = 1$$ 3. Equating the $z$-coordinates: $$3 = \frac{1 + b + 5}{3}$$ Multiply by 3: $$9 = b + 6 \implies b = 3$$ Compiling the values obtained: $a = 2$, $b = 3$, and $c = 1$. This corresponds to option (D).

Step 4: Final Answer:
The values of the parameters are $a = 2$, $b = 3$, $c = 1$, which matches option (D).