Point A divides segment BC in the ratio 4 : 1.Co - ordiantes of B are (6, 1) and C are `((7)/(2),6)`.What are the co-ordinates of point A ?

Options

A. `(4, 3)`
B. `(4, 5)`
C. `(2, 5)`
D. `(3,5)`

Answer

To find the coordinates of point A that divides the segment BC in the ratio 4:1, we can use the section formula. The section formula states that if a point A divides the line segment joining points B(x1, y1) and C(x2, y2) in the ratio m:n, then the coordinates of point A (x, y) can be calculated as follows:

\[ x = \frac{mx2 + nx1}{m+n} \]
\[ y = \frac{my2 + ny1}{m+n} \]

Given:
- B(6, 1) and C(7/2, 6)
- The ratio m:n = 4:1

### Step 1: Identify the coordinates of B and C
- B has coordinates (x1, y1) = (6, 1)
- C has coordinates (x2, y2) = (7/2, 6)

### Step 2: Assign the values of m and n
- m = 4
- n = 1

### Step 3: Calculate the x-coordinate of point A
Using the formula for x:
\[ x = \frac{4 \cdot \frac{7}{2} + 1 \cdot 6}{4 + 1} \]
Calculating the numerator:
\[ = \frac{4 \cdot \frac{7}{2} + 6}{5} \]
\[ = \frac{14 + 6}{5} = \frac{20}{5} = 4 \]

### Step 4: Calculate the y-coordinate of point A
Using the formula for y:
\[ y = \frac{4 \cdot 6 + 1 \cdot 1}{4 + 1} \]
Calculating the numerator:
\[ = \frac{24 + 1}{5} = \frac{25}{5} = 5 \]

### Step 5: Write the coordinates of point A
The coordinates of point A are (4, 5).

### Final Answer
The coordinates of point A are (4, 5).

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