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:
Given the centroid G(4,3,3) and vertices A(a,3,1), B(4,5,b), C(6,c,5), we need to find the values of a, b, and c.

Step 2: Key Formula or Approach:
The centroid coordinates are the arithmetic means of the corresponding vertex coordinates: x_G = (x₁+x₂+x₃)/3, and similarly for y and z.

Step 3: Detailed Explanation:
For x: 4 = (a+4+6)/3 → a = 2. For y: 3 = (3+5+c)/3 → c = 1. For z: 3 = (1+b+5)/3 → b = 3. Thus a=2, b=3, c=1.

Step 4: Final Answer:
The parameters are a=2, b=3, c=1, matching option (D).