Question

The ratio in which the point \[ (3,4) \] divides the line segment joining \[ (1,2) \] and \[ (5,6) \] is:

• A. \(1:1\)
• B. \(1:2\)
• C. \(2:1\)
• D. \(3:1\)

Hint:

If the coordinates of a point are exactly the averages of the corresponding coordinates of the endpoints, then the point is the midpoint and divides the segment in the ratio \(1:1\).

Answer 1

The Correct Option is A

Concept: The midpoint of a line segment joining two points \[ (x_1,y_1) \] and \[ (x_2,y_2) \] is given by \[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right). \] If a point coincides with the midpoint, then it divides the segment into two equal parts. Consequently, the ratio of division is \(1:1\).

Step 1: Identify the endpoints. The given endpoints are \[ (1,2) \] and \[ (5,6). \]

Step 2: Calculate the midpoint. Using the midpoint formula, \[ \left( \frac{1+5}{2}, \frac{2+6}{2} \right). \] \[ = \left( \frac6{2}, \frac8{2} \right). \] \[ = (3,4). \]

Step 3: Compare with the given point. The given point is \[ (3,4). \] The calculated midpoint is also \[ (3,4). \] Hence the given point is exactly the midpoint.

Step 4: Interpret geometrically. Since the point is the midpoint, it divides the line segment into two equal lengths. Therefore the ratio of division is \[ 1:1. \]

Step 5: Verification. Distance from \((1,2)\) to \((3,4)\) equals \[ \sqrt{(2)^2+(2)^2}. \] Distance from \((3,4)\) to \((5,6)\) is also \[ \sqrt{(2)^2+(2)^2}. \] Hence both segments are equal.

Step 6: Final Conclusion. \[ \boxed{1:1} \] Hence the correct answer is \[ \boxed{\text{Option (A)}}. \]