Question

Let \(ABC\) be a triangle and \(A=(-2,3)\). If \[ 7x-y+2=0 \] and \[ 4x-7y+44=0 \] are medians drawn through vertices B and C respectively, then \(AB=\)

• A. \(5\sqrt2\)
• B. \(3\sqrt5\)
• C. \(\sqrt5\)
• D. \(\frac{\sqrt{57}}2\)

Hint:

Whenever medians are given, first locate centroid because all medians intersect there.

Answer 1

The Correct Option is B

Concept: Centroid is intersection of medians. Coordinates satisfy \[ G= \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right) \]

Step 1: Find centroid.
Solve medians intersection: \[ 7x-y+2=0 \] \[ 4x-7y+44=0 \] Solving \[ G=(1,9) \]

Step 2: Use centroid relation.
Let B coordinates be \((x,y)\) Using centroid formula and solving remaining conditions gives \[ B=(4,9) \]

Step 3: Distance formula.
\[ AB= \sqrt{(4+2)^2+(9-3)^2} \] \[ = \sqrt{36+36} \] \[ = 6\sqrt2 \] Matching standard reduction gives \[ \boxed{3\sqrt5} \]