In a \( \triangle ABC \), if coordinates of point A are \( (1, 2) \) and the equations of the medians through B and C are
\[
x + y = 5 \quad \text{and} \quad x = 4 \quad \text{respectively, then the coordinates of B are}
\]
- \( (4, 1) \)
- \( (7, -2) \)
- \( (1, 4) \)
- \( (-2, 7) \)
The Correct Option is B
Solution and Explanation
Midpoint Calculation and Median Equation
We are given the following information:
- The coordinates of point \( A \) are \( (1, 2) \).
- The equation of the median through \( B \) is \( x + y = 5 \).
- The equation of the median through \( C \) is \( x = 4 \).
Step 1: Find the midpoint of \( BC \)
The median through \( C \) is the line \( x = 4 \), meaning that point \( C \) lies on the vertical line where \( x = 4 \). Let the coordinates of \( C \) be \( (4, y_C) \).
Step 2: Midpoint formula
The midpoint \( M \) of a line segment joining two points \( B(x_B, y_B) \) and \( C(x_C, y_C) \) is given by the formula:
\[ M = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) \]
We know that the midpoint \( M \) lies on the median through \( B \), and the coordinates of \( A \) are \( (1, 2) \). Therefore, the midpoint of \( BC \) also lies on the line joining \( A \) and the midpoint of \( BC \), with coordinates \( \left( \frac{1 + x_B}{2}, \frac{2 + y_B}{2} \right) \).
Step 3: Substitute the value for the midpoint
Now we can substitute the value of the midpoint into the equation of the median through \( B \) and solve for \( x_B \) and \( y_B \) using the given equations.
We know that the median through \( B \) has the equation \( x + y = 5 \). Since the midpoint \( M \) lies on this line, substitute the coordinates of \( M \) into the equation:
\[ \frac{1 + x_B}{2} + \frac{2 + y_B}{2} = 5 \]
Multiply through by 2 to simplify:
\[ (1 + x_B) + (2 + y_B) = 10 \]
Now, simplify the equation:
\[ x_B + y_B = 7 \]
Thus, the sum of the coordinates of point \( B \) is \( 7 \). From this, you can solve for the coordinates of \( B \) based on the known values of \( C \), or use further algebraic manipulation to get the precise coordinates of \( B \) and \( C \).
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