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

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.

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?

Can we apply the technique we used to find the distance between (2,2) and (-1, -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?

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).

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.

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