Problem 61 - Precalculus

Question

Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (3,2) && y=2 x+1\end{array}$$

Step-by-Step Solution

Answer

The distance between the point (3,2) and the line y = 2x+1 is \(\sqrt{5}\).

Step 1: Identify the Variables

First, identify the variables to be used in the distance formula. For the given line equation y = 2x + 1, it can be rewritten as 2x - y + 1 = 0. Therefore, A = 2, B = -1 and C = 1. And the point is given as (3,2), which means x1 = 3 and y1 = 2.

Step 2: Apply the Variables into Distance Formula

Next, substitute these variables into the distance formula. This results in \(D = \frac{|2*3 - 1*2 + 1|}{\sqrt{2^2 + (-1)^2}} = \frac{|6 - 2 + 1|}{\sqrt{4 + 1}} = \frac{5}{\sqrt{5}}\).

Step 3: Simplify the Result

Finally, simplify the result of the distance formula. This results in \(D = \frac{5}{\sqrt{5}} = \sqrt{5}\) after square root rationalization.

Key Concepts

Distance Formula

The distance between a point and a line can be calculated using the distance formula. This is a special formula used in geometry to measure how far a specific point is from a given line. When you have a point \(x_1, y_1\) and a line with the equation \ Ax + By + C = 0\, the distance \(D\) from the point to the line is given by:
\[D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\]This formula helps to determine how far off the point is relative to the line, using basic arithmetic operations and square roots. Remember:

• \(A\), \(B\), and \(C\) are from the line equation.
• \(x_1\), \(y_1\) are the coordinates of the point.
• Ensure the line is in the \(Ax + By + C = 0\) format before applying the formula.

Understanding this formula is essential for solving problems involving distances in geometry, particularly those concerning maps, navigation, and various design fields.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic processes to solve geometric problems. The coordinate plane, made up of the x-axis and y-axis, allows us to graph equations and study points, lines, and figures. This method simplifies computations that don't require scale drawings but use number-based measurements.
By using a Cartesian coordinate system, where every point is defined by an \(x\) and a \(y\) value, problems about lines and curves can be solved more easily.

• Each point is a pair of numerical coordinates: \( (x, y) \).
• Equations that describe lines or curves can be plotted, showing how they interact with each other.

Coordinate geometry transforms traditional geometry by allowing corrections and comparisons using mathematical formulas. It bridges the gap between algebra and geometry by representing geometric shapes as equations.

Line Equations

Line equations are essential in understanding the relationships between points on a coordinate plane. The general form of a line equation is written as \(Ax + By + C = 0\). However, lines commonly appear in other forms like the slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Each form of a line equation can help solve different types of problems in geometry:

• The general form \(Ax + By + C = 0\) is useful for algebraic manipulation and finding distances.
• The slope-intercept form \(y = mx + b\) easily shows the slope and position of the line on the graph.

When calculating distances, the line equation should ideally be rearranged to the general form to directly use in the distance formula. Understanding different forms of line equations is crucial for graph interpretation and making meaningful calculations in coordinate geometry.