Problem 68
Question
Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (2,1) && -2 x+y=2\end{array}$$
Step-by-Step Solution
Verified Answer
The distance between the point (2,1) and the line -2x+y=2 is \(\sqrt{5}\).
1Step 1: Identify the coefficients from the line equation
In the line equation -2x+y=2, the coefficients are \(A = -2, B = 1\), and \(C = -2\).
2Step 2: Insert the coefficients and point coordinates into the formula
Substitute A, B, C and the point coordinates (2,1) into the distance formula: \(d = \frac{{ |-2*2 + 1*1 -2| }}{{\sqrt{(-2)^2 + 1^2}}}\).
3Step 3: Simplify the fraction
Perform the calculations in the formula and simplify: \(d = \frac{{ |-4 + 1 - 2| }}{{\sqrt{4 + 1}}} = \frac{{ |-5| }}{{\sqrt{5}}} = \frac{5}{\sqrt{5}}\).
4Step 4: Simplify the Radical
Simplify the fraction by multiplying top and bottom by the square root of 5 \(d = \frac{5\sqrt{5}}{5} = \sqrt{5}\)
Key Concepts
Line Equation CoefficientsPoint CoordinatesDistance Formula
Line Equation Coefficients
Understanding the coefficients in a line equation is essential for solving various geometric problems, including finding the distance from a point to a line. A linear equation in two dimensions is typically written in the form Ax + By + C = 0, where A, B, and C are the line's coefficients.
These coefficients have geometric significance: A and B indicate the slope of the line, which is the rate at which the line rises or falls as it moves from left to right. Coefficient C represents the line’s intercept, which is where the line crosses the y-axis. In our exercise, the given line equation is -2x + y = 2. This means that A = -2, B = 1, and the constant C = -2. These coefficients will be used with the point's coordinates to find the shortest distance to the line.
These coefficients have geometric significance: A and B indicate the slope of the line, which is the rate at which the line rises or falls as it moves from left to right. Coefficient C represents the line’s intercept, which is where the line crosses the y-axis. In our exercise, the given line equation is -2x + y = 2. This means that A = -2, B = 1, and the constant C = -2. These coefficients will be used with the point's coordinates to find the shortest distance to the line.
Point Coordinates
Point coordinates are a fundamental concept of geometry, indicating a point's precise location on a plane. Coordinates are typically written as an ordered pair (x, y), where x represents the horizontal position, and y represents the vertical position relative to the origin (0, 0) of the coordinate system.
In the context of our problem, the coordinates of the point are (2, 1). This tells us that the point is located 2 units to the right of the origin along the x-axis, and 1 unit up the y-axis. To find its distance from a line, these coordinates must be plugged into the distance formula along with the line's equation coefficients.
In the context of our problem, the coordinates of the point are (2, 1). This tells us that the point is located 2 units to the right of the origin along the x-axis, and 1 unit up the y-axis. To find its distance from a line, these coordinates must be plugged into the distance formula along with the line's equation coefficients.
Distance Formula
The distance formula is a precious tool in geometry, allowing us to calculate the shortest distance between a point and a line. The formula is derived from the Pythagorean theorem and is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where d is the distance, A and B are coefficients from the line's equation, and (x_1, y_1) are the coordinates of the point. Absolute value is used to account for the distance being a positive quantity, regardless of the direction.
To apply the distance formula to our problem, we substitute the coefficients and point coordinates from our earlier steps: \[ d = \frac{|-2(2) + 1(1) - 2|}{\sqrt{(-2)^2 + 1^2}} \] This simplifies to \[ d = \frac{5}{\sqrt{5}} \] and then to d = \sqrt{5}, which gives us the distance between the point and the line.
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where d is the distance, A and B are coefficients from the line's equation, and (x_1, y_1) are the coordinates of the point. Absolute value is used to account for the distance being a positive quantity, regardless of the direction.
To apply the distance formula to our problem, we substitute the coefficients and point coordinates from our earlier steps: \[ d = \frac{|-2(2) + 1(1) - 2|}{\sqrt{(-2)^2 + 1^2}} \] This simplifies to \[ d = \frac{5}{\sqrt{5}} \] and then to d = \sqrt{5}, which gives us the distance between the point and the line.
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