Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-22: Prove that \(C\) is on the perpendicular bisector of segment \(A B\). $$ A(-3,2), \quad B(5,-4), \quad C(7,7) $$

Step-by-Step Solution

Answer

Point C is on the perpendicular bisector of segment AB.

Step 1: Find the midpoint of segment AB

To find the midpoint of segment \(AB\), use the midpoint formula: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \(A(-3,2)\) and \(B(5,-4)\) into this formula.\[ \text{Midpoint of } AB = \left(\frac{-3 + 5}{2}, \frac{2 + (-4)}{2}\right) = \left(1, -1\right) \]

Step 2: Calculate the slope of segment AB

The slope \(m\) of a line through 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 \(A(-3, 2)\) and \(B(5, -4)\):\[ m_{AB} = \frac{-4 - 2}{5 + 3} = \frac{-6}{8} = -\frac{3}{4} \]

Step 3: Determine the slope of the perpendicular bisector

The slope of a line perpendicular to another with slope \(m\) is the negative reciprocal. For \(m_{AB} = -\frac{3}{4}\), the perpendicular slope is:\[ m_{\perp} = \frac{4}{3} \]

Step 4: Write the equation of the perpendicular bisector

The equation of a line in slope-intercept form is \(y = mx + b\). Use the midpoint \((1, -1)\) and perpendicular slope \(\frac{4}{3}\) to find \(b\):\[ -1 = \frac{4}{3} \cdot 1 + b \]\[ b = -1 - \frac{4}{3} = -1 - \frac{4}{3} = -\frac{7}{3} \]So, the equation of the perpendicular bisector is:\[ y = \frac{4}{3}x - \frac{7}{3} \]

Step 5: Verify if point C lies on the perpendicular bisector

Substitute point \(C(7,7)\) into the equation of the perpendicular bisector to see if it satisfies the equation:\[ 7 = \frac{4}{3} \cdot 7 - \frac{7}{3} \]\[ 7 = \frac{28}{3} - \frac{7}{3} \]\[ 7 = \frac{21}{3} \]\[ 7 = 7 \]Since the equation holds true, point \(C\) is on the perpendicular bisector of \(AB\).

Key Concepts

Midpoint Formula

The midpoint formula is essential when trying to find the middle point of a line segment between two given points. It works by averaging the x-coordinates and y-coordinates of the endpoints. The formula is expressed as follows:

• In a coordinate plane, if you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula for the midpoint \(M\) is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).
• This concept simplifies locating the point exactly halfway between \((x_1, y_1)\) and \((x_2, y_2)\) on a graph.

In the problem, points \(A(-3, 2)\) and \(B(5, -4)\) were used in the formula, resulting in a midpoint of \((1, -1)\). This midpoint is crucial for establishing the perpendicular bisector that passes through this point.

Slope of a Line

The slope of a line represents its steepness and direction in the coordinate plane, and it's a fundamental element in mathematical geometry. Calculating the slope, often denoted as \(m\), requires two points on the line.

• The formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• This formula calculates rise \(y_2 - y_1\) over run \(x_2 - x_1\), quantifying how much the line goes up (or down) for each step it goes across.

In the exercise using points \(A(-3, 2)\) and \(B(5, -4)\), the calculated slope was \(m_{AB} = -\frac{3}{4}\), reflecting its negative direction.

Negative Reciprocal

The negative reciprocal of a slope is central when determining the slope of a line that is perpendicular to another. This relation is vital in geometry, particularly for constructs like perpendicular bisectors.

• To find the negative reciprocal, flip the fraction of the slope and change its sign.
• For example, if the slope \(m\) is \(-\frac{3}{4}\), its negative reciprocal becomes \(\frac{4}{3}\).
• This new slope is used to form a line that is entirely perpendicular to the original.

In the step-by-step solution, this was used to find the slope of the perpendicular bisector of segment \(AB\).

Equation of a Line

Finding the equation of a line requires both a point on the line and the line's slope. The standard form typically used is the slope-intercept form, \(y = mx + b\).

• Here, \(m\) is the line’s slope, and \(b\) is the y-intercept, where the line crosses the y-axis.
• Once the slope and a point on the line are known, solving for \(b\) can craft the full equation.

In the problem, the midpoint \((1, -1)\) and negative reciprocal slope \(\frac{4}{3}\) were used. By substituting into the slope-intercept equation and solving for \(b\), the complete equation of the perpendicular bisector was determined to be \(y = \frac{4}{3}x - \frac{7}{3}\).

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra and geometry using a coordinate plane to investigate geometric figures and their properties. It allows for the visualization and analysis of geometric concepts.

• By using numerical methods, it delivers precise calculations in geometric problems.
• In this exercise, it was used to coordinate geometry principles to establish and verify the position of point \(C(7, 7)\) on the perpendicular bisector of \(AB\).
• This involves applying coordinate geometry concepts like the midpoint formula, the equation of a line, and slopes.

The successful use of coordinate geometry verified that point \(C\) did indeed lie on the stated line, fulfilling the problem's requirements.