Key Concepts and Formulas

• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• Slope of a Line: The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2-y_1}{x_2-x_1}$. The slope of a line given by the equation $Ax+By+C=0$ is $m = -\frac{A}{B}$.
• Angle Between Two Lines: If two lines have slopes $m_1$ and $m_2$, then the tangent of the angle $\theta$ between them is given by $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$.
• Tangent Double Angle Formula: $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$.

Step-by-Step Solution

Step 1: Find the coordinates of vertex C

• Understanding the Median: The median through B connects B to the midpoint of AC. Let M be the midpoint of AC. We want to find the coordinates of C, which we will denote as $(x_C, y_C)$.
• Coordinates of M: Using the midpoint formula with A(-3, 1) and C($x_C, y_C$), the coordinates of M are: $M\left(\frac{x_C - 3}{2}, \frac{y_C + 1}{2}\right)$
• M lies on the Median: The equation of the median through B is $2x + y - 3 = 0$. Since M lies on the median, we substitute the coordinates of M into the equation: $2\left(\frac{x_C - 3}{2}\right) + \left(\frac{y_C + 1}{2}\right) - 3 = 0$ $x_C - 3 + \frac{y_C + 1}{2} - 3 = 0$ Multiplying by 2 to eliminate the fraction: $2x_C - 6 + y_C + 1 - 6 = 0$ $2x_C + y_C - 11 = 0$ $2x_C + y_C = 11 \quad \text{(Equation 1)}$
• C lies on the Angle Bisector: The equation of the angle bisector of C is $7x - 4y - 1 = 0$. Since C($x_C, y_C$) lies on this line, we substitute the coordinates of C into the equation: $7x_C - 4y_C - 1 = 0$ $7x_C - 4y_C = 1 \quad \text{(Equation 2)}$
• Solving for $x_C$ and $y_C$: We now have a system of two linear equations: $2x_C + y_C = 11 \quad \text{(Equation 1)}$ $7x_C - 4y_C = 1 \quad \text{(Equation 2)}$ From Equation 1, we have $y_C = 11 - 2x_C$. Substituting this into Equation 2: $7x_C - 4(11 - 2x_C) = 1$ $7x_C - 44 + 8x_C = 1$ $15x_C = 45$ $x_C = 3$ Substituting $x_C = 3$ back into $y_C = 11 - 2x_C$: $y_C = 11 - 2(3) = 11 - 6 = 5$ Therefore, the coordinates of vertex C are (3, 5).

Step 2: Calculate the slopes of AC and the angle bisector of C

• Slope of AC ($m_{AC}$): Using A(-3, 1) and C(3, 5): $m_{AC} = \frac{5 - 1}{3 - (-3)} = \frac{4}{6} = \frac{2}{3}$
• Slope of Angle Bisector ($m_{bisector}$): The equation of the angle bisector is $7x - 4y - 1 = 0$. We can rewrite this in slope-intercept form (y = mx + c) to find the slope: $4y = 7x - 1$ $y = \frac{7}{4}x - \frac{1}{4}$ So, the slope of the angle bisector is $m_{bisector} = \frac{7}{4}$.

Step 3: Calculate tan($\theta/2$)

• We know that the angle bisector divides the angle at C into two equal angles of $\theta/2$. Therefore, $\theta/2$ is the angle between the line AC and the angle bisector.
• Using the formula for the angle between two lines: $\tan\left(\frac{\theta}{2}\right) = \left|\frac{m_{AC} - m_{bisector}}{1 + m_{AC}m_{bisector}}\right|$ $\tan\left(\frac{\theta}{2}\right) = \left|\frac{\frac{2}{3} - \frac{7}{4}}{1 + \frac{2}{3} \cdot \frac{7}{4}}\right| = \left|\frac{\frac{8 - 21}{12}}{1 + \frac{14}{12}}\right| = \left|\frac{-\frac{13}{12}}{\frac{26}{12}}\right| = \left|-\frac{13}{26}\right| = \frac{1}{2}$

Step 4: Calculate tan($\theta$) using the double angle formula

• We want to find $\tan \theta$, and we know $\tan(\theta/2) = \frac{1}{2}$. Using the double angle formula: $\tan \theta = \frac{2 \tan(\theta/2)}{1 - \tan^2(\theta/2)} = \frac{2(\frac{1}{2})}{1 - (\frac{1}{2})^2} = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}$

Common Mistakes & Tips

• Sign Errors: Be careful with signs when calculating slopes and applying the angle between lines formula. Always double-check your calculations.
• Using the Correct Formula: Ensure you use the correct formula for the angle between two lines, and that you take the absolute value to find the acute angle.
• Solving System of Equations: When solving for $x_C$ and $y_C$, make sure to substitute correctly and avoid algebraic errors.

Summary

We first found the coordinates of vertex C by using the information about the median through B and the angle bisector of C. Then, we calculated the slopes of AC and the angle bisector, and used the angle between lines formula to find $\tan(\theta/2)$. Finally, we used the double angle formula to find $\tan \theta$, which resulted in $\frac{4}{3}$.

The final answer is \boxed{\frac{4}{3}}, which corresponds to option (C).