Coordinate Geometry: Distance Formula and Slope of a Line Quiz
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Questions and Answers

What does the distance formula enable us to find?

  • The distance between two points on a coordinate plane (correct)
  • The equation of a line
  • The slope of a line passing through two points
  • The area under a curve

    • Which theorem is the distance formula based on?

  • Law of Sines
  • Arc Length Theorem
  • Pythagorean Theorem (correct)
  • Triangle Inequality Theorem

    • What does the slope of a line measure?

  • Distance between two points
  • Angle of inclination of the line from the x-axis
  • Rate of change in the y-direction per unit increase in the x-direction (correct)
  • Area enclosed by the line and x-axis

    • If the slope of a line is 0, what can be said about the line?

    <p>It is horizontal</p> Signup and view all the answers

    For a horizontal line, what is the form of its equation?

    <p>$y = k$</p> Signup and view all the answers

    In the slope formula, what does $(y_2 - y_1)$ represent?

    <p>Change in $y$-direction</p> Signup and view all the answers

    What is the equation of a line if it is vertical?

    <p>x = k</p> Signup and view all the answers

    What is the main application of the distance formula in coordinate geometry?

    <p>Calculating distance between two points</p> Signup and view all the answers

    What does the slope formula help us find?

    <p>Slope of a line</p> Signup and view all the answers

    If the points are (2, 4) and (5, 7), what is the distance between them?

    <p>$3 ext{ units}$</p> Signup and view all the answers

    What is the slope of the line passing through (-1, 2) and (3, 7)?

    <p>$rac{5}{4}$</p> Signup and view all the answers

    In which situation would the slope be undefined?

    <p>When the line is vertical</p> Signup and view all the answers

    Study Notes

    Coordinate Geometry: Distance Formula and Slope of a Line

    In the realm of coordinate geometry, two fundamental concepts — distance formula and slope of a line — unlock the power to measure and analyze the behavior of points and lines on a plane. These ideas are based on the Cartesian coordinate system, where each point is defined by its (x)- and (y)-coordinates. Let's take a closer look at these two essential concepts.

    Distance Formula

    The distance formula is a fundamental tool in coordinate geometry that enables us to find the distance between any two points ((x_1, y_1)) and ((x_2, y_2)) on a coordinate plane.

    [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

    Here, (d) is the distance between the two points. This formula is based on the Pythagorean theorem, which states that the length of the hypotenuse of a right-angled triangle is the square root of the sum of the squares of the lengths of the other two sides.

    Slope of a Line

    The slope of a line is a measure of its steepness, giving us an indication of how much the line rises or falls in the (y)-direction for each unit increase in the (x)-direction. The slope (m) of a line that passes through points ((x_1, y_1)) and ((x_2, y_2)) is defined as follows:

    [ m = \frac{y_2 - y_1}{x_2 - x_1} ]

    If the line is horizontal (steepness is 0), then the slope is 0, and the equation of the line has the form (y = k) for some constant (k). If the line is vertical (steepness is undefined), then the slope is undefined, and the equation has the form (x = k) for some constant (k).

    Applications

    The distance formula and slope of a line have numerous applications in coordinate geometry, including:

    1. Calculating the distance between any two points on a coordinate plane.
    2. Determining whether or not two lines intersect, and if they do, finding the point of intersection.
    3. Locating the midpoint of a line segment connecting two points.
    4. Finding the equation of a line passing through two points.
    5. Solving equations involving lines and line segments.

    Example Problems

    1. Find the distance between the points ((2, 4)) and ((5, 7)).

    Using the distance formula, we have:

    [ d = \sqrt{(5 - 2)^2 + (7 - 4)^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \text{ units} ]

    1. Find the slope of the line passing through the points ((-1, 2)) and ((3, 7)).

    Using the slope formula, we have:

    [ m = \frac{7 - 2}{3 - (-1)} = \frac{5}{4} ]

    Conclusion

    The distance formula and slope of a line are two fundamental concepts in coordinate geometry that enable us to measure and analyze the behavior of points and lines on a plane. These ideas are essential in graphing, solving equations, and understanding the relationships between points and lines. By mastering these concepts, you will have a strong foundation for exploring the rich world of coordinate geometry.

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    Description

    Explore fundamental concepts in coordinate geometry such as the distance formula and slope of a line, which are essential for measuring and analyzing points and lines on a plane. Learn how to calculate distances between points and determine the steepness of lines based on their coordinates. Mastering these concepts will provide a strong foundation for understanding relationships between points and lines in a Cartesian coordinate system.

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