Question : Point A divides segment BC in the ratio 4 : 1. Coordinates of B are (6, 1) and coordinates of C are ($\frac{7}{2}$, $6$). What are the coordinates of point A?
Option 1: (4, 3)
Option 2: (4, 5)
Option 3: (2, 5)
Option 4: (3, 5)
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Correct Answer: (4, 5)
Solution :
The point A coordinates by using the section formula, $A(x,y)=(\frac{m\times x_2 + n\times x_1}{m+n},\frac{m\times y_2 + n \times y_1} {m+n})$ where $A(x_1,y_1)$ and $B(x_2,y_2)$ are the coordinates of the points which divides the line segment in the ratio $m:n$.
Given: $m=4$, $n=1$, $x_1=6, y_1=1, x_2= \frac{7}{2}, \text{and}\; y_2= 6$
From the section formula, the $x$ coordinate of A $=\frac{4\times \frac{7}{2}+1\times6 }{4+1}$
$=\frac{20}{5}=4$
Similarly, the y coordinate of A is $\frac{4\times 6+1\times1}{4+1}$.
$=\frac{25}{5}=5$
The coordinates of point A are (4, 5).
Hence, the correct answer is (4, 5).
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