8.4 - The Distance Formula

 Introduction:

This is a set of guiding questions and materials for creating your own lesson plan on introducing the distance formula.

 

 Common Core State Standards:

  • 8.G.8
  • G-GPE.4
  • G-GPE.5

 

 Learning Objectives:

  • To realize the need of having an embedding structure (a metric) to be able to measure distance.
  • To realize that the choice of a metric is arbitrary.
  • To understand how to derive the distance formula as an application of the Pythagorean Theorem.
  • To apply the distance formula to exercises and problems.
  • To apply the distance formula to derive the equation of a circle centered at the origin.

 

 Guiding Questions:

Activity: Draw two points on the board.

  • How far apart are the points?
  • What do we need to determine or measure that distance?
    • A ruler or a coordinate system
    • Both embed the system with a structure to be able to measure.
  • How far are the points (2,2) and (-1, -2)?
    • Elicit from students the construction of a right triangle to solve it.
  • What do we need to calculate the distance between the two points in the activity above?
  • What can we call the points?
    • Suggest P1 and P2.
  • How can we embed a coordinate system?
  • What are the coordinates of P1 and P2?
    • Suggest P1 = (x1, y1) and P2 = (x2, y2).
  • Can we apply the technique we used to find the distance between (2,2) and (-1, -2)?
    • What are the coordinates of the point at vertex of the right angle?
      • (x2, y1) if x1 < x2 and y1 < y2
      • What is the length of the legs of the right triangle?
        • (x2 – x1) and (y2 – y1).
        • What is the distance between the two points?
          • d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}d=√(x2−x1)2+(y2−y1)2
          • Given that a circle is the set of all points that are equidistant from a point called the center, what equation would represent a circle centered at the origin?

           

           Notes for Teachers:

          • I have indented some suggestions in the questions above.
          • If students easily grasp the concept of the equation of a circle centered at the origin, you may want to try the general case for a circle centered at the point (h, k). If you do so, point out the difference between parameters (in this case, h and k) and variables (x and y).

           

           Video of the Day:

          8.4.1 The Distance Formula and the Equation of a Circle Centered at the Origin

          • We show the standard derivation of the distance formula and connect it to the equation of a circle of radius r centered at the origin.
          8.4.1 The Distance Formula.mp4 Download 8.4.1 The Distance Formula.mp4Play media comment.

           

           Exercises and Problems:

          • Successful Mathematics: North Carolina Common Core Math 1 Item Bank, Nagy-Lup, p. 90, problems 37-40.
            • 37. Check if the triangle with the vertices A(4, 2), B(7, -1), and C(11, 3) is a right angle triangle. Explain your reasoning.
            • 38. Check if the triangle with the vertices A(1, 3), B(3, 1), and C(5, 5) is an isosceles triangle. Explain your reasoning.
            • 39. Check if the point lies on the circle centered at the origin and containing (0, 2). Explain your reasoning.
            • 40. Check if the quadrilateral with the vertices A(3, 1), B(5, -3), C(9, -1), and D (7, 3) is a rectangle. Explain your reasoning.
          • Khan Academy

           Online Sources:

           

           Additional Resource for Teachers:

          If you wish to download the contents of this page as a printable pdf, click here:  Download 8.4 Distance Formula.pdf