Key Concepts and Formulas

• The orthocenter of a triangle is the point of intersection of its altitudes.
• The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• If two lines are perpendicular, the product of their slopes is -1. If line 1 has slope $m_1$, a line perpendicular to it has slope $m_2 = -\frac{1}{m_1}$.
• The point-slope form of a line passing through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$.

Step-by-Step Solution

Given the vertices $A(3,-7)$, $B(-1,2)$, and $C(4,5)$. We want to find the orthocenter $(\alpha, \beta)$.

Step 1: Find the equation of the altitude AD (from A to BC)

Our goal is to find the equation of the line through A that is perpendicular to BC.

• 1.1 Calculate the slope of BC ($m_{BC}$): Using points $B(-1,2)$ and $C(4,5)$: $m_{BC} = \frac{5-2}{4-(-1)} = \frac{3}{5}$ Explanation: This slope is needed to find the slope of the perpendicular altitude.

• 1.2 Calculate the slope of AD ($m_{AD}$): Since AD is perpendicular to BC: $m_{AD} = -\frac{1}{m_{BC}} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3}$ Explanation: The negative reciprocal gives the perpendicular slope.

• 1.3 Find the equation of AD: Using point-slope form with $A(3,-7)$ and $m_{AD} = -\frac{5}{3}$: $y - (-7) = -\frac{5}{3}(x - 3)$ $y + 7 = -\frac{5}{3}(x - 3)$ Multiply by 3: $3y + 21 = -5x + 15$ $5x + 3y + 6 = 0 \quad \text{(Equation 1)}$ Explanation: This is the equation of the altitude AD.

Step 2: Find the equation of the altitude BE (from B to AC)

Now, we find the equation of the line through B that is perpendicular to AC.

• 2.1 Calculate the slope of AC ($m_{AC}$): Using points $A(3,-7)$ and $C(4,5)$: $m_{AC} = \frac{5 - (-7)}{4 - 3} = \frac{12}{1} = 12$ Explanation: This slope is needed to find the slope of the perpendicular altitude.

• 2.2 Calculate the slope of BE ($m_{BE}$): Since BE is perpendicular to AC: $m_{BE} = -\frac{1}{m_{AC}} = -\frac{1}{12}$ Explanation: Taking the negative reciprocal again.

• 2.3 Find the equation of BE: Using point-slope form with $B(-1,2)$ and $m_{BE} = -\frac{1}{12}$: $y - 2 = -\frac{1}{12}(x - (-1))$ $y - 2 = -\frac{1}{12}(x + 1)$ Multiply by 12: $12y - 24 = -x - 1$ $x + 12y - 23 = 0 \quad \text{(Equation 2)}$ Explanation: This is the equation of the altitude BE.

Step 3: Find the Orthocenter $(\alpha, \beta)$ by Solving the System of Altitude Equations

We need to solve the system:

1. $5x + 3y = -6$
2. $x + 12y = 23$

Solve for $x$ in Equation 2: $x = 23 - 12y$ Explanation: This allows us to use substitution.

Substitute into Equation 1: $5(23 - 12y) + 3y = -6$ $115 - 60y + 3y = -6$ $115 - 57y = -6$ $-57y = -121$ $y = \frac{121}{57}$ So, $\beta = \frac{121}{57}$. Explanation: We found the y-coordinate of the orthocenter.

Substitute $y$ back into the expression for $x$: $x = 23 - 12\left(\frac{121}{57}\right)$ $x = 23 - \frac{1452}{57}$ $x = \frac{23 \times 57 - 1452}{57} = \frac{1311 - 1452}{57} = \frac{-141}{57}$ Simplify by dividing by 3: $x = \frac{-47}{19}$ So, $\alpha = -\frac{47}{19}$. Explanation: We found the x-coordinate.

The orthocenter is $\left(-\frac{47}{19}, \frac{121}{57}\right)$.

**Step 4: Evaluate the Expression $9\alpha - 6\beta + 60$}

Substitute the values of $\alpha$ and $\beta$: $9\alpha - 6\beta + 60 = 9\left(-\frac{47}{19}\right) - 6\left(\frac{121}{57}\right) + 60$ $= -\frac{423}{19} - \frac{726}{57} + 60$ $= -\frac{423}{19} - \frac{242}{19} + 60$ $= \frac{-423 - 242}{19} + 60$ $= \frac{-665}{19} + 60$ $= -35 + 60 = 25$ Explanation: Calculating the final value.

Common Mistakes & Tips

• Double-check the signs when calculating slopes and substituting values into equations.
• Be careful when working with fractions. Ensure you are finding common denominators and simplifying where possible.
• Remember to take the negative reciprocal when finding the slope of a perpendicular line.

Summary

We found the orthocenter of the triangle by finding the equations of two altitudes and solving the system of equations. We then evaluated the given expression using the coordinates of the orthocenter.

The final answer is $\boxed{25}$, which corresponds to option (C).