Problem 26 - The Calculus with Analytic Geometry

Question

Find equations of the perpendicular bisectors of the sides of the triangle having vertices \(A(-1,-3), B(5,-3)\), and \(C(5,5)\), and prove that they meet in a point.

Step-by-Step Solution

Answer

The equations are \(x = 2\), \(y = 1\), and \(y - 1 = -\frac{3}{4}(x - 2)\). They intersect at (2, 1).

Step 1 - Find the Midpoints

Calculate the midpoints of each side of the triangle.Midpoint of AB: \ \(\text{Midpoint}_{AB} = \left( \frac{-1+5}{2}, \frac{-3+(-3)}{2}\right) = (2, -3) \)\Midpoint of BC: \ \(\text{Midpoint}_{BC} = \left( \frac{5+5}{2}, \frac{-3+5}{2}\right) = (5, 1) \)\Midpoint of AC: \ \(\text{Midpoint}_{AC} = \left( \frac{-1+5}{2}, \frac{-3+5}{2}\right) = (2, 1) \)

Step 2 - Find the Slopes of Each Side

Calculate the slopes of each side of the triangle.\Slope of AB: \ \(m_{AB} = \frac{-3 - (-3)}{5 - (-1)} = 0\)\Slope of BC: \ \(m_{BC} = \frac{5 - (-3)}{5 - 5} = \text{undefined}\)\Slope of AC: \ \(m_{AC} = \frac{5 - (-3)}{5 - (-1)} = \frac{8}{6} = \frac{4}{3}\)

Step 3 - Calculate the Slopes of the Perpendicular Bisectors

Use the negative reciprocal of each side's slope to find the slope of the perpendicular bisector.\Perpendicular bisector of AB: \ \(m_{P_{AB}} = \text{undefined}\) (since the slope of AB is 0, the perpendicular bisector is a vertical line).\Perpendicular bisector of BC: \ \(m_{P_{BC}} = 0\) (since the slope of BC is undefined, the perpendicular bisector is a horizontal line).\Perpendicular bisector of AC: \ \(m_{P_{AC}} = -\frac{3}{4}\)

Step 4 - Determine the Equations of Perpendicular Bisectors

Use the point-slope form to write the equations of the perpendicular bisectors.\Perpendicular bisector of AB through (2, -3): \ \(x = 2\)\Perpendicular bisector of BC through (5, 1): \ \(y = 1\)\Perpendicular bisector of AC through (2, 1): \ \(y - 1 = -\frac{3}{4}(x - 2)\)

Step 5 - Find the Intersection Point

Solve for the intersection point of the equations of the perpendicular bisectors.\Intersection of \(x = 2\) and \(y = 1\) gives the point (2, 1).\Verify with the third perpendicular bisector: \(y - 1 = -\frac{3}{4}(2 - 2)\) gives \(y = 1\), so (2, 1) is indeed the intersection point.

Key Concepts

Midpoint Calculation

Calculating the midpoint of a line segment helps you find the exact center between two points. The formula to find the midpoint is:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]where \(x_1, y_1\) and \(x_2, y_2\) are the endpoints of the line segment. Let's use this formula to find the midpoints of the sides of our triangle.
- For side AB with points A(-1, -3) and B(5, -3), the midpoint is:\[\left( \frac{-1 + 5}{2}, \frac{-3 + (-3)}{2} \right) = (2, -3)\]- For side BC with points B(5, -3) and C(5, 5), the midpoint is:\[\left( \frac{5 + 5}{2}, \frac{-3 + 5}{2} \right) = (5, 1)\]- For side AC with points A(-1, -3) and C(5, 5), the midpoint is:\[\left( \frac{-1 + 5}{2}, \frac{-3 + 5}{2} \right) = (2, 1)\]Finding the midpoints is the first step in determining the perpendicular bisectors.

Slope of a Line

The slope of a line indicates its steepness and direction. It is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]where \(x_1, y_1\) and \(x_2, y_2\) are points on the line. Let's find the slopes of each side of the triangle.
- For side AB with points A(-1, -3) and B(5, -3), the slope is:\[ m_{AB} = \frac{-3 - (-3)}{5 - (-1)} = 0\]This slope of 0 indicates a horizontal line.
-For side BC with points B(5, -3) and C(5, 5), the slope is undefined since the denominator is zero:\[ m_{BC} = \frac{5 - (-3)}{5 - 5} = \text{undefined}\]This indicates a vertical line.
-For side AC with points A(-1, -3) and C(5, 5), the slope is:\[ m_{AC} = \frac{5 - (-3)}{5 - (-1)} = \frac{8}{6} = \frac{4}{3}\]Calculating these slopes helps us find the slopes of the perpendicular bisectors.

Point-Slope Form

The point-slope form of a line equation helps you write the equation of a line when you know the slope and a point on the line.The formula is:\[ y - y_1 = m(x - x_1) \]where \(x_1, y_1\) is a point on the line and \(m\) is the slope. We can use this form to write the equations of the perpendicular bisectors once we've calculated their slopes.
-For the perpendicular bisector of AB, the slope is undefined indicating a vertical line, so the equation is \(x = 2\).-For the perpendicular bisector of BC, the slope is 0 indicating a horizontal line, so the equation is \(y = 1\).-For the perpendicular bisector of AC with slope \(-\frac{3}{4}\), the equation through midpoint (2,1) is:\[ y - 1 = -\frac{3}{4}(x - 2)\]Using these equations, we can then determine the intersection point of the bisectors and show that they meet at a common point, proving that they are indeed the perpendicular bisectors.