Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Find equations for the altitudes of the triangle with vertices \(A(-3,2), B(5,4)\), and \(C(3,-8)\), and find the point at which the altitudes intersect.

Step-by-Step Solution

Answer

Altitudes are: \(y = -4x + 4\), \(y = -\frac{1}{6}x + \frac{1}{2}\), \(y = \frac{3}{5}x + \frac{1}{5}\). Intersection is \((\frac{21}{23}, \frac{8}{23})\)."

Step 1: Find the Slope of Each Side

First, for each pair of triangle vertices, calculate the slope of the side connecting them. For side \(AB\):\[m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{5 + 3} = \frac{2}{8} = \frac{1}{4}\]For side \(BC\):\[m_{BC} = \frac{-8 - 4}{3 - 5} = \frac{-12}{-2} = 6\]For side \(AC\):\[m_{AC} = \frac{-8 - 2}{3 + 3} = \frac{-10}{6} = -\frac{5}{3}\]

Step 2: Find the Perpendicular Slopes

To find the equation of the altitude from a vertex to the opposite side, calculate the perpendicular slope for each side's slope.For altitude from \(C\) to \(AB\): Perpendicular to \(m_{AB} = \frac{1}{4}\) is \(-4\).For altitude from \(A\) to \(BC\): Perpendicular to \(m_{BC} = 6\) is \(-\frac{1}{6}\).For altitude from \(B\) to \(AC\): Perpendicular to \(m_{AC} = -\frac{5}{3}\) is \(\frac{3}{5}\).

Step 3: Write the Equations of the Altitudes

Using the point-slope formula \(y - y_1 = m(x - x_1)\), write equations for each altitude.Altitude from \(C\) to \(AB\) (through point \(C(3,-8)\) with slope \(-4\)):\[y + 8 = -4(x - 3) \Rightarrow y = -4x + 4\]Altitude from \(A\) to \(BC\) (through point \(A(-3,2)\) with slope \(-\frac{1}{6}\)):\[y - 2 = -\frac{1}{6}(x + 3) \Rightarrow y = -\frac{1}{6}x + \frac{1}{2}\]Altitude from \(B\) to \(AC\) (through point \(B(5,4)\) with slope \(\frac{3}{5}\)):\[y - 4 = \frac{3}{5}(x - 5) \Rightarrow y = \frac{3}{5}x + \frac{1}{5}\]

Step 4: Find the Intersection of the Altitudes

Solve the system of equations to find the intersection point. We can solve it step by step, using pairs of equations.First, set the two equations for lines from steps 3 equal to find their intersection:\(-4x + 4 = -\frac{1}{6}x + \frac{1}{2}\)Solve for \(x\):\[-4x + \frac{1}{6}x = \frac{1}{2} - 4\]\[-\frac{24x}{6} + \frac{x}{6} = -\frac{7}{2}\]\[-24x + x = -21\]\[-23x = -21\]\[x = \frac{21}{23}\]Substitute \(x\) back into one of the original equations, such as \(y = -4x + 4\):\[y = -4\left(\frac{21}{23}\right) + 4\]\[y = -\frac{84}{23} + \frac{92}{23}\]\[y = \frac{8}{23}\]This gives us point \( (\frac{21}{23}, \frac{8}{23}) \), where two altitudes intersect.

Key Concepts

Equations of Lines

The equation of a line is a fundamental concept in mathematics, providing a way to describe a straight line in the Cartesian coordinate system. In two dimensions, lines are often expressed via the slope-intercept form:

• If we know the slope (\(m\)) and y-intercept (\(b\)), we write the line equation as \(y = mx + b\).
• This formula directly tells you the slope of the line and where it crosses the y-axis.
• Another common form is the point-slope formula, which is especially handy for finding the equation of a line when a point on the line and the slope is known: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line.

Both these forms play a crucial role in writing the equations of altitudes in a triangle. For each altitude, we start by identifying the slope of the relevant side of the triangle and then find the perpendicular slope to help write its equation. Once we have a slope and a point (the vertex of the triangle from which the altitude is drawn), the point-slope form is used to derive the altitude's equation.

Intersection of Lines

The intersection of two lines is the point at which they meet on the plane. When dealing with altitudes of a triangle, the idea is to find where these lines intersect, which is called the orthocenter of the triangle. To determine the intersection point:

• We need the equations of the two lines involved.
• Set the equations equal to each other to solve for \(x\)
• Substitute the found \(x\) value back into one of the line equations to determine \(y\).

For lines in the form \(y = mx + b\), equating them allows us to solve for a common \(x\). Once \(x\) is found, it is plugged back into either line equation to find the \(y\)-coordinate. This process identifies the exact coordinates where the two lines intersect. Such intersections are critical in various applications, such as finding the meeting point of altitudes in triangles, which confirms their concurrent nature.

Perpendicular Slopes

Perpendicular slopes are critical when working with geometric figures, like triangles, where understanding orthogonality is necessary. The slope of a line is a measure of its steepness, and for two lines to be perpendicular:

• Their slopes must be negative reciprocals of each other.
• This means if one line has a slope of \(m\), the perpendicular line will have a slope of \(-\frac{1}{m}\).

This relationship is essential in deriving equations for altitudes of triangles, as these altitudes are perpendicular to the sides of the triangle. Thus, knowing the slope of a triangle's side allows us to easily compute the slope of the perpendicular altitude. This calculation is a preliminary step before using the point-slope formula to write the equation of the altitude and, subsequently, determine where it intersects with others.