Question

Let \[ O(0,0,0) \] and \[ A(2,1,-3) \] be vertices of a triangle \(OAB\). If \[ (-1,2,1) \] is the midpoint of side \(AB\), and the perimeter of the triangle is \[ \sqrt2\,(k+l\sqrt7+m\sqrt{13}), \] then the value of \[ k+l+m \] is:

• A. \(7\)
• B. \(8\)
• C. \(5\)
• D. \(6\)

Hint:

In three-dimensional geometry, if a midpoint and one endpoint are known, always find the missing endpoint first. This usually converts a difficult coordinate geometry problem into a straightforward distance-formula calculation.

Answer 1

The Correct Option is B

Concept: The midpoint formula in three dimensions is \[ M\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right). \] If the midpoint and one endpoint are known, the coordinates of the other endpoint can be found easily. The distance between two points in three-dimensional geometry is \[ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. \] After obtaining all three side lengths, we add them to obtain the perimeter.

Step 1: Determine the coordinates of \(B\). Given midpoint \[ M(-1,2,1). \] Using midpoint formula, \[ \left( \frac{2+x_B}{2}, \frac{1+y_B}{2}, \frac{-3+z_B}{2} \right) = (-1,2,1). \] Comparing coordinates, \[ 2+x_B=-2 \Rightarrow x_B=-4, \] \[ 1+y_B=4 \Rightarrow y_B=3, \] \[ -3+z_B=2 \Rightarrow z_B=5. \] Hence \[ B=(-4,3,5). \]

Step 2: Calculate \(OA\). \[ OA = \sqrt{2^2+1^2+(-3)^2}. \] \[ = \sqrt{4+1+9}. \] \[ = \sqrt{14} = \sqrt2\,\sqrt7. \]

Step 3: Calculate \(OB\). \[ OB = \sqrt{(-4)^2+3^2+5^2}. \] \[ = \sqrt{16+9+25}. \] \[ = \sqrt{50} = 5\sqrt2. \]

Step 4: Calculate \(AB\). \[ AB = \sqrt{(-6)^2+2^2+8^2}. \] \[ = \sqrt{36+4+64}. \] \[ = \sqrt{104}. \] \[ = 2\sqrt{26} = 2\sqrt2\,\sqrt{13}. \]

Step 5: Find the perimeter. \[ P = OA+OB+AB. \] \[ = \sqrt2\sqrt7 + 5\sqrt2 + 2\sqrt2\sqrt{13}. \] Factor \(\sqrt2\), \[ P = \sqrt2 \left( 5+\sqrt7+2\sqrt{13} \right). \] Comparing with \[ \sqrt2 \left( k+l\sqrt7+m\sqrt{13} \right), \] we get \[ k=5, \qquad l=1, \qquad m=2. \]

Step 6: Compute the required value. \[ k+l+m = 5+1+2. \] \[ =8. \]

Step 7: Final Conclusion. \[ \boxed{8} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]