Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-24: Find a formula that expresses the fact that an arbitrary point \(P(x, y)\) is on the perpendicular bisector \(l\) of segment \(A B\). $$ A(-4,-3), B(6,1) $$

Step-by-Step Solution

Answer

The formula for point \(P(x, y)\) on the perpendicular bisector is \(y = -\frac{5}{2}x + \frac{3}{2}\).

Step 1: Find the Midpoint of Segment AB

The midpoint of segment \(AB\) is calculated using the midpoint formula, \(M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For points \(A(-4,-3)\) and \(B(6,1)\), the coordinates of the midpoint are: \[ M\left( \frac{-4 + 6}{2}, \frac{-3 + 1}{2} \right) = M(1,-1) \]

Step 2: Determine the Slope of AB

The slope of line segment \(AB\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting in points \(A(-4,-3)\) and \(B(6,1)\), we find: \[ m = \frac{1 - (-3)}{6 - (-4)} = \frac{4}{10} = \frac{2}{5} \]

Step 3: Calculate the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector \(l\), which is the negative reciprocal of the slope of \(AB\), is calculated as follows: \[ m_l = -\frac{1}{\left(\frac{2}{5}\right)} = -\frac{5}{2} \]

Step 4: Write the Equation of the Perpendicular Bisector

Using the point-slope form \(y - y_1 = m(x - x_1)\) and the slope found in Step 3. With point \(M(1,-1)\), we get: \[ y + 1 = -\frac{5}{2}(x - 1) \] Simplifying, the equation becomes: \[ y = -\frac{5}{2}x + \frac{3}{2} \]

Step 5: Test if Point P(x, y) Falls on Line l

To confirm that an arbitrary point \(P(x, y)\) lies on \(l\), substitute the values \(x\) and \(y\) into the equation found in Step 4: \[ y = -\frac{5}{2}x + \frac{3}{2} \] If both sides of the equation are equal, \(P(x, y)\) lies on the perpendicular bisector.

Key Concepts

Midpoint Formula

Finding the midpoint of a line segment is a key concept in geometry. The midpoint formula allows us to find the exact center point between two endpoints, providing a way to divide the segment into two equal parts.
In our example, the endpoints are given by point A (-4, -3) and point B (6, 1).
To find the midpoint, apply the formula:

• For the x-coordinate: take the average of the x-values from both points, which is \((\frac{-4 + 6}{2})\).
• For the y-coordinate: take the average of the y-values from both points, which is \((\frac{-3 + 1}{2})\).

Together, these give the midpoint as M(1, -1), effectively splitting the segment AB perfectly in half.
The midpoint is particularly useful when dealing with perpendicular bisectors, as the bisector will always pass through this midpoint.

Slope of a Line

The slope of a line expresses its steepness and direction. It is crucial in analytic geometry for understanding the relationship between two points.
Given two points, A (-4, -3) and B (6, 1), the slope is determined by the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Inserting our specific values results in:

• \[ m = \frac{1 - (-3)}{6 - (-4)} = \frac{4}{10} = \frac{2}{5} \]

This indicates the line rises \(2\) units for every \(5\) units it runs horizontally.
Understanding the slope helps in deriving the perpendicular bisector, as the slope of the bisector is the negative reciprocal of this slope.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, involves using algebraic equations to represent geometric shapes and solve geometric problems.
It provides tools for understanding and visualizing geometric figures in a coordinate plane. By expressing points with coordinates, we can define lines, angles, and distances between them using equations.
For instance, when we have points A and B given by coordinates, we can calculate their properties, such as the midpoint and slope, and consequently derive the equation for the perpendicular bisector.
In this exercise, we've seen how analytical geometry allows us to systematically create the equation for the perpendicular bisector, aiding in more advanced geometric problem-solving without traditional compass-and-straightedge methods.

Point-Slope Form

The point-slope form of a line is a crucial tool in linear equations, especially when working with specific slopes and points.
It is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.
For our perpendicular bisector, with a slope of \(-\frac{5}{2}\) and passing through point M(1, -1), the point-slope form becomes:

• \[ y + 1 = -\frac{5}{2}(x - 1) \]

This equation helps in confirming whether any point \((x, y)\) lies on this line.
By simplifying the equation, we make it easier to identify points that reside on the line, transforming the equation into the form \(y = -\frac{5}{2}x + \frac{3}{2}\).
Understanding this form is essential for solving problems involving linear equations and bisectors in analytical geometry.