Key Concepts and Formulas

• Isosceles Triangle Property: In an isosceles triangle, the altitude from the vertex angle to the base bisects the base. This means the median from vertex A to the midpoint of BC is perpendicular to BC.
• Centroid Formula: The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$.
• Perpendicular Lines: If two lines are perpendicular, the product of their slopes is -1.

Step-by-Step Solution

Step 1: Finding the Coordinates of Vertex B

• Why this step? Vertex B lies on both the line BC and the given line x + 3y = 7. Solving these equations simultaneously will give the coordinates of B.
• Working: The equation of line BC is $2x + y = 4$ (Equation 1). The equation of the other line is $x + 3y = 7$ (Equation 2). From Equation 1, we have $y = 4 - 2x$. Substituting this into Equation 2 gives: $x + 3(4 - 2x) = 7$ $x + 12 - 6x = 7$ $-5x = -5$ $x = 1$ Substituting $x = 1$ back into $y = 4 - 2x$, we get: $y = 4 - 2(1) = 2$
• Result: The coordinates of vertex B are (1, 2).

Step 2: Finding the Midpoint M of the Base BC

• Why this step? Since $\triangle ABC$ is isosceles with AB = AC, the median AM is perpendicular to BC. We can find the equation of AM and then find M, the intersection of AM and BC.
• Working: The slope of BC is found from the equation $2x + y = 4$, which can be rewritten as $y = -2x + 4$. The slope of BC, $m_{BC}$, is -2. Since AM is perpendicular to BC, the slope of AM, $m_{AM}$, is the negative reciprocal of $m_{BC}$: $m_{AM} = -\frac{1}{-2} = \frac{1}{2}$. The equation of line AM passing through A(6, 1) with slope $\frac{1}{2}$ is given by: $y - 1 = \frac{1}{2}(x - 6)$ $2y - 2 = x - 6$ $x - 2y = 4$ (Equation 3) To find the coordinates of M, we solve the system of equations formed by Equation 1 ($2x + y = 4$) and Equation 3 ($x - 2y = 4$). From Equation 3, $x = 2y + 4$. Substituting into Equation 1: $2(2y + 4) + y = 4$ $4y + 8 + y = 4$ $5y = -4$ $y = -\frac{4}{5}$ Substituting $y = -\frac{4}{5}$ back into $x = 2y + 4$: $x = 2\left(-\frac{4}{5}\right) + 4 = -\frac{8}{5} + \frac{20}{5} = \frac{12}{5}$
• Result: The coordinates of the midpoint M are $\left(\frac{12}{5}, -\frac{4}{5}\right)$.

Step 3: Finding the Coordinates of Vertex C

• Why this step? M is the midpoint of BC. Knowing B and M, we can use the midpoint formula to find C.
• Working: Let $C = (x_C, y_C)$. We have $B = (1, 2)$ and $M = \left(\frac{12}{5}, -\frac{4}{5}\right)$. Using the midpoint formula: $\frac{x_B + x_C}{2} = x_M \implies \frac{1 + x_C}{2} = \frac{12}{5}$ $1 + x_C = \frac{24}{5}$ $x_C = \frac{24}{5} - 1 = \frac{19}{5}$ $\frac{y_B + y_C}{2} = y_M \implies \frac{2 + y_C}{2} = -\frac{4}{5}$ $2 + y_C = -\frac{8}{5}$ $y_C = -\frac{8}{5} - 2 = -\frac{18}{5}$
• Result: The coordinates of vertex C are $\left(\frac{19}{5}, -\frac{18}{5}\right)$.

Step 4: Calculating the Centroid ($\alpha$, $\beta$)

• Why this step? We now have all three vertices: $A(6, 1)$, $B(1, 2)$, and $C\left(\frac{19}{5}, -\frac{18}{5}\right)$. We can directly apply the centroid formula.
• Working: $\alpha = \frac{x_A + x_B + x_C}{3} = \frac{6 + 1 + \frac{19}{5}}{3} = \frac{\frac{35 + 5 + 19}{5}}{3} = \frac{\frac{54}{5}}{3} = \frac{18}{5}$ $\beta = \frac{y_A + y_B + y_C}{3} = \frac{1 + 2 - \frac{18}{5}}{3} = \frac{\frac{5 + 10 - 18}{5}}{3} = \frac{-\frac{3}{5}}{3} = -\frac{1}{5}$
• Result: The centroid $(\alpha, \beta)$ is $\left(\frac{18}{5}, -\frac{1}{5}\right)$.

Step 5: Final Calculation of $15(\alpha + \beta)$

• Why this step? This is the final value requested by the problem.
• Working: $\alpha + \beta = \frac{18}{5} - \frac{1}{5} = \frac{17}{5}$ $15(\alpha + \beta) = 15 \times \frac{17}{5} = 3 \times 17 = 51$
• Result: $15(\alpha + \beta) = 51$.

Common Mistakes & Tips

• Using Distance Formula: Trying to find C directly using the distance formula AB = AC can lead to more complex calculations. Using the perpendicularity of the median is more efficient.
• Sign Errors: Pay close attention to signs when calculating slopes and using the point-slope form of a line.
• Fraction Arithmetic: Be careful when adding and subtracting fractions. Double-check your work.

Summary

This problem tests understanding of coordinate geometry, including finding the intersection of lines, using the midpoint formula, and applying properties of isosceles triangles. By systematically finding the vertices and then the centroid, we found that $15(\alpha + \beta) = 51$.

The final answer is $\boxed{51}$, which corresponds to option (C).