Select All The Points That Are On The Line Through (0 5) And (2 8) A.(5 11) B.(5 10) C.(6 14) D.(30 50)

Select All The Points That Are On The Line Through (0 5) And (2 8) A.(5 11) B.(5 10) C.(6 14) D.(30 50)

Understanding how to determine whether specific points lie on a line passing through two given points is a fundamental concept in coordinate geometry. This article provides a comprehensive guide to identifying points that lie on the line passing through points (0, 5) and (2, 8). We will explore the mathematical principles involved, step-by-step procedures, and apply these to evaluate options A, B, C, and D. Whether you're a student preparing for exams or a math enthusiast, this guide aims to clarify the process and enhance your problem-solving skills.

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Understanding the Problem: Points and Lines in Coordinate Geometry

Before diving into calculations, it's essential to understand the core concepts involved:

What Is a Line in Coordinate Geometry?

• A line in a two-dimensional plane is a straight, infinitely extending path connecting points.
• It can be represented algebraically using equations such as the slope-intercept form (y = mx + b) or point-slope form.

What Does It Mean for a Point to Lie on a Line?

• A point (x, y) lies on a line if substituting its coordinates into the line's equation satisfies the equation.
• In simpler terms, the point's coordinates satisfy the mathematical relationship defining the line.

Why Is It Important to Find Points on a Line?

• Understanding which points lie on a particular line is crucial in many areas, including graphing, solving geometric problems, and analyzing data trends.

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Step-by-Step Guide to Find the Equation of the Line Through (0, 5) and (2, 8)

To determine whether given points are on the line passing through (0, 5) and (2, 8), we first need the equation of that line.

Step 1: Calculate the Slope (m)

• The slope of a line passing through two points \((x1, y1)\) and \((x2, y2)\) is given by:

\[ m = \frac{y2 - y1}{x2 - x1} \]

• For points (0, 5) and (2, 8):

\[ m = \frac{8 - 5}{2 - 0} = \frac{3}{2} \]

Step 2: Write the Equation Using Point-Slope Form

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

\[ y - y1 = m(x - x1) \]

• Using point (0, 5):

\[ y - 5 = \frac{3}{2}(x - 0) \Rightarrow y - 5 = \frac{3}{2}x \]

• Simplify to slope-intercept form:

\[ y = \frac{3}{2}x + 5 \]

Result: The equation of the line passing through (0, 5) and (2, 8):

\[
\boxed{y = \frac{3}{2}x + 5}
\]

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Evaluating Points A, B, C, and D for Line Inclusion

Now that we have the line's equation, the next step is to check each given point to see if it satisfies the equation.

Method for Verification

• For each point \((x, y)\), substitute the x-coordinate into the line's equation.
• If the resulting y-value matches the y-coordinate of the point, then the point lies on the line.
• Otherwise, it does not.

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Analysis of Each Point

Point A: (5, 11)

• Substitute \(x = 5\):

\[ y = \frac{3}{2} \times 5 + 5 = \frac{3}{2} \times 5 + 5 = \frac{15}{2} + 5 = 7.5 + 5 = 12.5 \]

• The point's y-coordinate is 11, but the calculated y-value is 12.5.

Conclusion: (5, 11) does not lie on the line.

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Point B: (5, 10)

• Substitute \(x = 5\):

\[ y = 7.5 \quad (\text{from previous calculation}) \]

• The point's y-coordinate is 10, which does not match 7.5.

Conclusion: (5, 10) does not lie on the line.

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Point C: (6, 14)

• Substitute \(x = 6\):

\[ y = \frac{3}{2} \times 6 + 5 = 9 + 5 = 14 \]

• The point's y-coordinate is 14, which matches the calculated y-value.

Conclusion: (6, 14) lies on the line.

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Point D: (30, 50)

• Substitute \(x = 30\):

\[ y = \frac{3}{2} \times 30 + 5 = 45 + 5 = 50 \]

• The point's y-coordinate is 50, which matches the calculated y-value.

Conclusion: (30, 50) lies on the line.

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Final Summary: Which Points Are On the Line?

Based on the calculations:

• Points on the line:


• (6, 14)


• (30, 50)


• Points not on the line:


• (5, 11)


• (5, 10)

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Implications and Applications

Understanding how to verify whether points lie on a specific line has broad applications, including:

Graphing and Visualization

• Plotting points accurately requires knowing whether they align with a given line.
• Helps in visual data analysis to identify trends.

Solving Geometric Problems

• Determines collinearity of points.
• Essential in proofs and geometric construction.

Data Analysis and Regression

• Verifying whether data points follow a linear trend.
• Critical in statistical modeling.

Educational Context

• A foundational skill in high school and college-level mathematics courses.

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Additional Tips for Line and Point Analysis

• Always calculate the line's equation carefully and verify each point systematically.
• Remember that floating-point inaccuracies can occur; use exact fractions when possible.
• For vertical lines (where \(x\) is constant), check if the x-coordinate matches the line's x-value.

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Conclusion

Determining whether specific points lie on a line passing through two given points involves deriving the line's equation and verifying each point through substitution. In this case, the line passing through (0, 5) and (2, 8) has the equation \(y = \frac{3}{2}x + 5\). Applying this to options given:

• Points (6, 14) and (30, 50) satisfy the equation.


• Points (5, 11) and (5, 10) do not.

This process underscores the importance of understanding the fundamentals of slope calculation, equation derivation, and point validation in coordinate geometry. Mastery of these concepts enhances problem-solving efficiency and accuracy, which is invaluable in academic and real-world applications.

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Keywords: coordinate geometry, line equation, slope calculation, points on a line, line passing through two points, verifying points, mathematical analysis, slope-intercept form, geometry problems, linear equations