On A Coordinate Plane, A Line Goes Through (0, 3) And (3, Negative 1). A Point Is At (negative 3, 2).What

Understanding the Coordinates and Their Significance

On A Coordinate Plane, A Line Goes Through (0, 3) And (3, Negative 1). A Point Is At (−3, 2). What does this information tell us about the relationship between these points and the line? In this article, we will explore the fundamentals of coordinate geometry, analyze the given points, and determine what the question might be asking. By understanding how to interpret the points and the line, students can enhance their skills in graphing, calculating slopes, and finding equations of lines.

Basics of the Coordinate Plane

What Is a Coordinate Plane?

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It allows us to plot points, lines, and shapes using pairs of numbers known as coordinates.

Understanding Coordinates

• Each point on the plane is represented by an ordered pair (x, y).
• The first number, x, indicates the position along the horizontal axis.
• The second number, y, indicates the position along the vertical axis.
• For example, the point (0, 3) is located at 0 on the x-axis and 3 on the y-axis.

Analyzing the Given Points and Line

The Known Points

• Point A: (0, 3)
• Point B: (3, -1)
• Point C: (−3, 2) — the point in question

Plotting the Points

Visualizing these points on the coordinate plane can help us better understand their relationships. Point A at (0, 3) is on the y-axis, three units above the origin. Point B at (3, -1) lies three units to the right and one unit below the x-axis. Point C at (−3, 2) is three units left of the origin and two units above the x-axis.

Calculating the Slope of the Line Through (0, 3) and (3, -1)

What Is Slope?

The slope of a line measures its steepness and is calculated as the change in y divided by the change in x between two points. The slope formula is:

slope (m) = (y₂ − y₁) / (x₂ − x₁)

Applying the Formula

Using the points (0, 3) and (3, -1):

• y₂ = -1, y₁ = 3
• x₂ = 3, x₁ = 0

Calculate the slope:

m = (−1 − 3) / (3 − 0) = (−4) / 3 = -4/3

Interpretation of the Slope

The slope of -4/3 indicates that for every 3 units the line moves horizontally to the right, it moves 4 units downward vertically.

Finding the Equation of the Line

Using Point-Slope Form

The point-slope form of a line’s equation is:

y − y₁ = m(x − x₁)

Choosing point (0, 3):

y − 3 = -4/3(x − 0)

Simplifies to:

y = -4/3 x + 3

Equation of the Line

• The line passing through (0, 3) and (3, -1) has the equation:
• y = -4/3 x + 3

Determining the Position of the Point (−3, 2)

Is the Point On the Line?

To check whether (−3, 2) lies on the line, substitute x = -3 into the line’s equation:

y = -4/3 (−3) + 3 = (4) + 3 = 7

Since the calculated y-value is 7, which does not match the point’s y-coordinate of 2, the point (−3, 2) does not lie on the line.

What Is the Relative Position?

• The point’s y-coordinate (2) is less than 7, the y-value on the line at x = -3.
• This means (−3, 2) is below the line at that x-value.

Interpreting the Question

Possible Questions Based on the Given Data

Given the information, the question might be asking one of the following:

• What is the equation of the line passing through the points (0, 3) and (3, -1)?
• Is the point (−3, 2) on, above, or below the line?
• What is the shortest distance from the point (−3, 2) to the line?
• Find the foot of the perpendicular from (−3, 2) to the line.

Calculating the Distance from the Point to the Line

Why Find the Distance?

Determining how far the point (−3, 2) is from the line can help in various applications, such as shortest path calculations or geometric proofs.

Distance Formula from a Point to a Line

The distance (d) from a point (x₀, y₀) to a line Ax + By + C = 0 is given by:

d = |A x₀ + B y₀ + C| / √(A² + B²)

Expressing the Line in Standard Form

The line's equation:

y = -4/3 x + 3

Multiply both sides by 3 to clear fractions:

3 y = -4 x + 9

Rearranged to standard form:

4 x + 3 y - 9 = 0

Calculating the Distance

Substitute (x₀, y₀) = (−3, 2):

d = |4(−3) + 3(2) − 9| / √(4² + 3²)
= |−12 + 6 − 9| / √(16 + 9)
= |−15| / √25
= 15 / 5
= 3

Thus, the point (−3, 2) is 3 units away from the line.

Summary and Practical Applications

Key Takeaways

• The line passing through (0, 3) and (3, -1) has the equation y = -4/3 x + 3.
• The point (−3, 2) lies below the line at that x-value, with a perpendicular distance of 3 units.
• Understanding how to find slopes, equations, and distances is crucial in coordinate geometry.

Real-World Uses of Coordinate Geometry

• Designing graphical interfaces and computer graphics
• Navigation and pathfinding
• Engineering and architectural design
• Physics for analyzing trajectories and forces

Conclusion

Analyzing points and lines on a coordinate plane combines algebraic techniques with geometric understanding. By calculating slopes, equations, and distances, students can interpret spatial relationships effectively. Whether the question is about the position of a point relative to a line or finding the shortest distance, mastering these concepts is fundamental in mathematics and its applications.