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Chapter 4: Problem 46

Graph each line. Construct a perpendicular segment through the given point. Then find the distance from the point to the line. $$y=x+2,(2,-2)$$

Short Answer

Expert verified
The distance from the point (2,-2) to the line y = x + 2 is \( 3\sqrt{2} \).

Step by step solution

01

Graph the Line

Start by graphing the line given by the equation \( y = x + 2 \). This is a linear equation in slope-intercept form, \( y = mx + b \), where \( m = 1 \) is the slope and \( b = 2 \) is the y-intercept. **Plot the y-intercept (0,2)** on the coordinate plane.Using the slope 1, move up 1 unit and to the right 1 unit to plot another point. Connect these points with a straight line.
02

Identify the Perpendicular Slope

To find the slope of a line that is perpendicular to the original line, use the negative reciprocal of the original slope. The original slope is 1, so the negative reciprocal is \( -1 \).
03

Equation of the Perpendicular Line

Use the point-slope form of a line equation \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (2, -2) \) and the slope \( m = -1 \). Substituting these values gives:\[ y + 2 = -1(x - 2) \]Simplify this equation to find the line equation:\[ y = -x + 0 \]
04

Graph the Perpendicular Line

Graph the line \( y = -x \) using the point \( (2, -2) \). From this point, use the slope \( -1 \) (move down 1 unit and to the right 1 unit, or up 1 unit and to the left 1 unit to plot another point). Draw the line through these points, ensuring it crosses the original line at a right angle.
05

Find Intersection Point

Find the intersection of the line \( y = x + 2 \) and the perpendicular line \( y = -x \). Set the equations equal to find the x-coordinate:\[ x + 2 = -x \]Solve for \( x \):\[ 2x = -2 \rightarrow x = -1 \]Substitute \( x = -1 \) back into one of the original equations to find \( y \):\[ y = -(-1) = 1 \]Thus, the intersection point is \( (-1, 1) \).
06

Calculate the Distance

Use the distance formula to find the distance from the point \( (2, -2) \) to the intersection point \( (-1, 1) \):\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute \( (x_1, y_1) = (2, -2) \) and \( (x_2, y_2) = (-1, 1) \):\[ d = \sqrt{(-1 - 2)^2 + (1 + 2)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \].
07

State the Distance

The distance from the point \( (2, -2) \) to the line \( y = x + 2 \) is \( 3\sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a common way to express the equation of a straight line. It is given by the formula: \[ y = mx + b \]. Here, \( m \) stands for the slope of the line, while \( b \) represents the y-intercept. This form is very handy because it directly tells us how to graph the line.
  • Start at the y-intercept \((0, b)\).
  • Then, use the slope \( m \) to plot further points. If \( m \) is 1, it indicates that for every unit you move to the right on the x-axis, you also move up by 1 unit on the y-axis. This creates a 45-degree diagonal line.
This understanding helps us determine how steep the line is and where it crosses the y-axis. For instance, for the equation \( y = x + 2 \), the line crosses the y-axis at \( (0, 2) \) and has a slope of 1.
Distance Formula
In coordinate geometry, the distance formula is used to calculate the distance between two points on a plane. The formula is derived from the Pythagorean theorem and is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
  • \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
  • This formula ensures that you can find the straight-line distance irrespective of the alignment of the points.
For example, to find the distance between the points \((2, -2)\) and \((-1, 1)\), simply substitute these values into the formula, simplify, and you get a distance of \( 3\sqrt{2} \). This distance is the straight-line measurement directly between these two points.
Point-Slope Form
The point-slope form is especially useful for writing the equation of a line when you know one point on the line and its slope. The general expression of the point-slope form is: \[ y - y_1 = m(x - x_1) \], where:
  • \((x_1, y_1)\) is a known point on the line.
  • \( m \) represents the slope.
To illustrate, consider using it to find the equation of a line perpendicular to another. If the original line has a slope of 1, the perpendicular line will have the negative reciprocal slope, which is \(-1\). Using the point \((2, -2)\), you substitute into the form to get \[ y + 2 = -1(x - 2) \], which simplifies to \[ y = -x \], representing the perpendicular line.
Coordinate Geometry
Coordinate Geometry involves using algebraic principles to understand geometric concepts on a coordinate plane. This branch of geometry allows us to visualize geometrical shapes using numbers and equations.
  • The fundamental elements include points, lines, and planes, which are translated into coordinates and algebraic equations.
  • It enables us to find slopes, distances, midpoints, and areas.
Using coordinate geometry, solving problems becomes systematic and logical. For instance, finding the intersection point of two lines, such as the original line \( y = x + 2 \) and the perpendicular \( y = -x \), involves setting the equations equal to each other and solving for the intercept. This intersection, \((-1, 1)\), is crucial as it represents where two geometrical paths cross on a plane.

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Most popular questions from this chapter

The total distance from Charlotte to Raleigh to Winston-Salem and back to Charlotte is about 292 miles. The distance from Charlotte to Winston-Salem is 22 miles less than the distance from Raleigh to Winston-Salem. The distance from Charlotte to Raleigh is 60 miles greater than the distance from Winston- Salem to Charlotte. Classify the triangle that connects Charlotte, Raleigh, and Winston-Salem. (IMAGE CAN'T COPY)

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