Find the ratio in which the point (5,4) divides the line joining points (2,1) and (7,6)

Answer

To find the ratio in which the point (5, 4) divides the line segment joining the points (2, 1) and (7, 6), we will use the section formula. The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P can be calculated using the following formulas:

\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 1: Identify the points
Let A = (2, 1) and B = (7, 6). The point P = (5, 4) divides the line segment AB.

### Step 2: Assume the ratio
Assume that point P divides the line segment AB in the ratio k:1. Thus, we can set m = k and n = 1.

### Step 3: Set up the equations using the section formula
Using the x-coordinates:
\[ 5 = \frac{k \cdot 7 + 1 \cdot 2}{k + 1} \]

Using the y-coordinates:
\[ 4 = \frac{k \cdot 6 + 1 \cdot 1}{k + 1} \]

### Step 4: Solve the x-coordinate equation
From the x-coordinate equation:
\[ 5(k + 1) = 7k + 2 \]
Expanding this gives:
\[ 5k + 5 = 7k + 2 \]
Rearranging the terms:
\[ 5 - 2 = 7k - 5k \]
\[ 3 = 2k \]
Thus,
\[ k = \frac{3}{2} \]

### Step 5: Solve the y-coordinate equation (optional for verification)
From the y-coordinate equation:
\[ 4(k + 1) = 6k + 1 \]
Expanding this gives:
\[ 4k + 4 = 6k + 1 \]
Rearranging the terms:
\[ 4 - 1 = 6k - 4k \]
\[ 3 = 2k \]
Thus,
\[ k = \frac{3}{2} \]

### Step 6: Write the final ratio
Since we assumed the ratio as k:1, we have:
\[ \text{Ratio} = \frac{3}{2} : 1 = 3 : 2 \]

### Final Answer
The point (5, 4) divides the line segment joining (2, 1) and (7, 6) in the ratio **3:2**.

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