Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-22: Prove that \(C\) is on the perpendicular bisector of segment \(A B\). $$ A(-4,-3), \quad B(6,1), \quad C(5,-11) $$

Step-by-Step Solution

Answer

Point C is on the perpendicular bisector of AB.

Step 1: Find the Midpoint of Segment AB

The midpoint of a segment can be found using the formula for the midpoint between two points, \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute the coordinates of points \( A(-4, -3) \) and \( B(6, 1) \) into this formula:\[ M_{AB} = \left( \frac{-4 + 6}{2}, \frac{-3 + 1}{2} \right) = (1, -1) \].

Step 2: Calculate the Slope of AB

The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using points \( A(-4, -3) \) and \( B(6, 1) \):\[ m_{AB} = \frac{1 - (-3)}{6 - (-4)} = \frac{4}{10} = 0.4 \].

Step 3: Determine the Slope of the Perpendicular Bisector

The perpendicular slope to a line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular bisector for line segment \( AB \) is:\[ m_{perpendicular} = -\frac{1}{m_{AB}} = -\frac{1}{0.4} = -2.5 \].

Step 4: Equation of Perpendicular Bisector

The equation of a line with slope \( m \) through a point \( (x_1, y_1) \) is:\[ y - y_1 = m(x - x_1) \]. Using the midpoint \((1, -1)\) and the perpendicular slope \(-2.5\):\[ y + 1 = -2.5(x - 1) \]Simplify this to:\[ y = -2.5x + 2.5 - 1 = -2.5x + 1.5 \].

Step 5: Check if C Lies on the Perpendicular Bisector

Substitute point \( C(5, -11) \) into the equation derived for the perpendicular bisector:\[ y = -2.5x + 1.5 \]Substitute and simplify:\[ -11 = -2.5(5) + 1.5 \]\[ -11 = -12.5 + 1.5 \]\[ -11 = -11 \].The equation holds true, which means point \( C \) lies on the perpendicular bisector of segment \( AB \).

Key Concepts

Midpoint Formula

The midpoint formula is a simple yet powerful tool in geometry, used to find the exact middle point between two endpoints of a line segment. Imagine you have two points, say \((x_1, y_1)\) and \((x_2, y_2)\). The midpoint \(M\) is calculated with the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This midpoint represents the average of the \(x\)-coordinates and \(y\)-coordinates of the two endpoints.

• It's like finding the center of a line segment.
• Very useful in proving lines bisect each other or in various geometric constructions.

In our exercise, using points \(A(-4,-3)\) and \(B(6,1)\), we find the midpoint to be \(M_{AB} = (1, -1)\). This midpoint is vital for determining characteristics like the perpendicular bisector of the segment AB.

Slope of a Line

Understanding the slope of a line helps unravel the incline or steepness between two points on a graph. The slope of a line is defined by the difference in the \(y\)-coordinates over the difference in the \(x\)-coordinates, given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]The slope not only indicates whether the line rises or falls, but here’s a brief breakdown of what it means:

• If \(m > 0\), the line ascends as you move left to right.
• If \(m < 0\), the line descends.
• If \(m = 0\), you have a horizontal line.
• When the slope is undefined, the line is vertical.

In our example, for line AB between points \((-4, -3)\) and \((6, 1)\), we calculated the slope \(m_{AB} = 0.4\). This suggests the line rises gently from \(A\) to \(B\). For perpendicular bisectors, the line's slope is crucial, as it will determine the negative reciprocal slope needed for the bisector.

Equation of a Line

When you know the slope of a line and have a point on that line, you can find its equation. This equation describes each point on the line as a set of \(x\) and \(y\) coordinates. The equation of a line with slope \(m\) going through a point \((x_1, y_1)\) is:\[y - y_1 = m(x - x_1)\]With some rearrangement and simplification, you'll often see it in the slope-intercept form: \[y = mx + b\]where \(b\) is the y-intercept.

• This form allows easy plotting by identifying slope \(m\) and intercept \(b\).
• Useful in determining whether a certain point lies on the line.

In our exercise, the equation of the perpendicular bisector was derived using midpoint \(M(1, -1)\) and the perpendicular slope \(m = -2.5\). The equation \(y = -2.5x + 1.5\) was then used to confirm that point \(C(5, -11)\) lies on this bisector. Remember, the specific line equation serves as a guide to checking alignment and position on graphs, confirming geometric relations.