Question

In a triangle \( ABC \), the coordinates of the vertex \( A \) are \( (1, 2) \). Equations of the median through \( B \) and \( C \) are respectively \( x + y = 5 \) and \( x = 4 \). Then the equation of side \( AB \) is:

• A. \( 2x - 3y + 4 = 0 \)
• B. \( 2x + 3y = 8 \)
• C. \( 3x - 2y = 0 \)
• D. \( 3x + 2y = 5 \)

Hint:

To find the equation of a line between two points, first find the slope using \( m = \frac{y_2 - y_1}{x_2 - x_1} \), then use the point-slope form to get the equation of the line.

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We are given the coordinates of vertex \( A(1, 2) \) and the equations of two medians. We need to find the equation of the side \( AB \). The median through \( B \) passes through the midpoint of \( AC \) and the median through \( C \) passes through the midpoint of \( AB \).

Step 2: Find the coordinates of point \( B \).
The median through \( C \) passes through the midpoint of \( AB \). Since the equation of this median is \( x = 4 \), the x-coordinate of point \( B \) is 7. Using the equation of the median through \( B \), we find that the y-coordinate of point \( B \) is -2. Therefore, the coordinates of point \( B \) are \( (7, -2) \).

Step 3: Calculate the equation of line \( AB \).
The coordinates of \( A(1, 2) \) and \( B(7, -2) \) are known. The slope of the line is \( m = -\frac{2}{3} \). The equation of line \( AB \) is \( y - 2 = -\frac{2}{3}(x - 1) \). Expanding and simplifying, we get the equation of line \( AB \) as \( 2x + 3y = 8 \).

Step 4: Conclusion.
Thus, the equation of side \( AB \) is \( 2x + 3y = 8 \). Therefore, the correct answer is option (B).