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

Step 1: Set up the problem.
We want the ratio in which the point $(3,4)$ splits the segment joining $(1,2)$ and $(5,6)$.

Step 2: Recall the section formula.
If a point divides the join of $(x_1,y_1)$ and $(x_2,y_2)$ in ratio $k:1$, its coordinates are $\left(\dfrac{kx_2+x_1}{k+1},\dfrac{ky_2+y_1}{k+1}\right)$.

Step 3: Use the $x$ coordinate.
Set the $x$ part equal to $3$. \[ \frac{k\cdot5+1}{k+1}=3. \]

Step 4: Solve for $k$.
Cross multiply: $5k+1=3(k+1)=3k+3$. So $5k-3k=3-1$, giving $2k=2$, hence $k=1$.

Step 5: Check with the $y$ coordinate.
With $k=1$, $\dfrac{1\cdot6+2}{1+1}=\dfrac{8}{2}=4$, which matches the given $y=4$.

Step 6: Write the ratio.
Since $k=1$, the ratio is $k:1=1:1$. The point is exactly the midpoint. \[ \boxed{1:1} \]