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Magazine Chapter 3: Problem 95
Find an equation of the perpendicular bisector of the line segment whose endpoints are given. \((2,3) ;(-4,7)\)
Short Answer
Expert verified
The equation is \(y = \frac{3}{2}x + \frac{13}{2}\).
Step by step solution
01
Calculate the Midpoint
To find the midpoint of the line segment given the endpoints \((2, 3)\) and \((-4, 7)\), use the midpoint formula: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). Plugging in the values: \(M = \left(\frac{2 + (-4)}{2}, \frac{3 + 7}{2}\right) = (-1, 5)\).
02
Determine the Slope of the Line Segment
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(m = \frac{y_2-y_1}{x_2-x_1}\). Substitute the coordinates of the endpoints: \(m = \frac{7-3}{-4-2} = \frac{4}{-6} = -\frac{2}{3}\).
03
Calculate the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the original line's slope. If the original slope is \(-\frac{2}{3}\), the perpendicular slope, \(m_p\), is \(\frac{3}{2}\).
04
Write the Equation of the Perpendicular Bisector
Using the point-slope form of a line's equation: \(y - y_1 = m(x - x_1)\), and the midpoint \((-1, 5)\) and slope \(\frac{3}{2}\), the equation is: \(y - 5 = \frac{3}{2}(x + 1)\). Converting to slope-intercept form \((y = mx + b)\), distribute and simplify: \(y - 5 = \frac{3}{2}x + \frac{3}{2}\), thus \(y = \frac{3}{2}x + \frac{3}{2} + 5\). Simplifying further gives \(y = \frac{3}{2}x + \frac{13}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Formula
The midpoint formula is a mathematical tool used to find the exact middle point between two given points on a coordinate plane. This is particularly useful for dividing a line segment into two equal halves. The formula is expressed as:\[M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\]Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two endpoints of the line segment. To implement this formula, simply add the \(x\)-coordinates of the two points together and divide by two to get the \(x\)-coordinate of the midpoint. Repeat this process for the \(y\)-coordinates to obtain the \(y\)-coordinate. This gives you the midpoint \((-1, 5)\) in our example.
- This formula is essential in geometry for finding centers of segments.
- It is a straightforward averaging process but a fundamental concept in coordinate geometry.
Slope of a Line
The slope of a line is a value that describes the tilt or steepness of the line on a graph. 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. The slope is essentially a ratio that compares the vertical change (rise) to the horizontal change (run) between two points. In our original exercise, the line segment's slope comes out to be \(-\frac{2}{3}\). This tells us that for every 3 units we move horizontally, the line descends 2 units vertically.
- A positive slope means the line is increasing; a negative slope means it's decreasing.
- The larger the absolute value, the steeper the line is.
Point-Slope Form
The point-slope form is a way of writing the equation of a line. It is useful for quickly forming an equation when a point on the line and the line's slope are known. The formula is:\[y - y_1 = m(x - x_1)\]In this formula, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope of the line. In our example, we use the midpoint as \((-1, 5)\) and the slope of the perpendicular bisector as \(\frac{3}{2}\). This results in:\[y - 5 = \frac{3}{2}(x + 1)\]
- It's valuable for lines where you know a coordinate and the slope.
- This form makes it easy to quickly adjust the slope or point to find a new line.
Slope-Intercept Form
The slope-intercept form is the most popular method of expressing the equation of a line. It is typically written as:\[y = mx + b\]Where \(m\) represents the slope, and \(b\) represents the y-intercept (where the line crosses the y-axis). From our exercise, we used the point-slope form to re-arrive at this by simplifying:\[y = \frac{3}{2}x + \frac{13}{2}\]This shows that the line crosses the y-axis at \(\frac{13}{2}\), and has a slope \(\frac{3}{2}\). This form easily allows us to understand how changes in \(x\) affect \(y\), and where the line begins when \(x = 0\).
- Ideal for graphing because it clearly shows the starting position.
- Helps in comparing different lines and understanding their slopes and intercepts directly.
