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=√(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.
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:
- Khan Academy
Additional Resource for Teachers:
If you wish to download the contents of this page as a printable pdf, click here: 8.4 Distance Formula.pdf
Download 8.4 Distance Formula.pdf