Question

Let the centroid of a triangle formed by the points \(A(4,x,1)\), \(B(y,-5,2)\), and \(C(7,8,3)\) be \(G(3,5,2)\), and \(CG\) meet \(AB\) in \(F\). Then \(F=\)

• A. \(\left(\frac{5}{2},\frac{3}{2},\frac{5}{2}\right)\)
• B. \(\left(\frac{11}{2},10,2\right)\)
• C. \(\left(1,\frac{7}{2},\frac{3}{2}\right)\)
• D. \((10,12,5)\)

Hint:

The centroid of a triangle lies on each median and divides the median in the ratio \(2:1\) from the vertex. Hence, the line joining a vertex and centroid meets the opposite side at its midpoint.

Answer 1

The Correct Option is C

Step 1: Use the formula of centroid.
The centroid of a triangle with vertices \[ A(x_1,y_1,z_1),\quad B(x_2,y_2,z_2),\quad C(x_3,y_3,z_3) \] is \[ \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\right) \] Here, \[ A(4,x,1),\quad B(y,-5,2),\quad C(7,8,3) \] and \[ G(3,5,2) \]

Step 2: Find \(x\) and \(y\).
Using the \(x\)-coordinate of centroid, \[ \frac{4+y+7}{3}=3 \] \[ 11+y=9 \] \[ y=-2 \] Using the \(y\)-coordinate of centroid, \[ \frac{x-5+8}{3}=5 \] \[ x+3=15 \] \[ x=12 \] So, \[ A(4,12,1) \] and \[ B(-2,-5,2) \]

Step 3: Understand the position of \(F\).
Since \(G\) is the centroid, it lies on the median from \(C\) to side \(AB\).
The line \(CG\) meets \(AB\) at \(F\).
Therefore, \(F\) is the midpoint of \(AB\).

Step 4: Find the midpoint of \(AB\).
Using midpoint formula, \[ F=\left(\frac{4+(-2)}{2},\frac{12+(-5)}{2},\frac{1+2}{2}\right) \] \[ F=\left(\frac{2}{2},\frac{7}{2},\frac{3}{2}\right) \] \[ F=\left(1,\frac{7}{2},\frac{3}{2}\right) \]

Step 5: Final conclusion.
Hence, \[ \boxed{\left(1,\frac{7}{2},\frac{3}{2}\right)} \]