## Using Coordinates To Prove Theorems

#### Aligned To Common Core Standard:

**Expressing Properties** - HSG-GPE.B.4

How to use Coordinates To Prove Theorems?
Coordinate geometry helps in proving a lot of theorems in mathematics. The proof usually begins with assigning of variables to the coordinates. There are several formulae in coordinate geometry that assist us in proving simple as well as complex theorems.
The formulae that help in this situation includes;
Distance Formula= √ ((x_{2} - x_{1}) ^{2} + (y_{2} - y_{1})^{2} )
The following formula helps in determining the length between two coordinate points.
Midpoint formula=((x_{1} + x_{2})/2,(y_{1} + y_{2})/2)
The following formula helps in finding the midpoint of a line connecting two coordinate points.
Slope Formula=(y_{2} - y_{1})/(x_{2} - x_{1} )
The formula helps in determining the slope of a line segment.
These formula can help in proving that a quadrilateral is a parallelogram, rectangle, or square. Moreover, it even helps in proving whether a triangle is a right-angled triangle or not. Just using the set of these formulae, you can prove all types of theorems! Students can use these lessons and worksheets to learn how to write coordinate proofs in a wide variety of situations.

### Printable Worksheets And Lessons

- Points and Circles
Step-by-step Lesson - Does this point reside on the circle?

- Guided Lesson
- Notice that you can prove the fact or disprove it.

- Guided Lesson Explanation
- The first question is a lengthy, but the others are quite manageable.

- Practice Worksheet
- These all follow a basic template, follow the means to get it
under control.

- Matching Worksheet
- "Yes" means you have or can proved it and "No"
means that you have or can disprove it.

- Equidistant from Two Intersecting
Lines Worksheet Five Pack - Some really unique critical thinking
questions for you.

- Equidistant from Two Parallel
Lines Worksheet Five Pack - More deep thinking questions for
you.

- Equidistant from Two Points
Worksheet Five Pack - If the ends were to meet in the middle,
where would that be?

#### Homework Sheets

I really find these problems difficult for most students. You need a solid understanding of trig.

- Homework 1 - We have start with the definition of a circle. A circle is the set of points that are an equidistant from the center.
- Homework 2 - Find how off they are from the center and go from there.
- Homework 3 - The coordinates make it much easier for all of you to do.

#### Practice Worksheets

Some of the problems might frustrate students. I would recommend completely the first problem with the students.

- Practice 1 - Prove or disprove that the point (-7.2, -5.1) lies on the circle centered at (-11.1, -7.5), and containing the point (-5.1, -2.2).
- Practice 2 - If we prove the points are the same distance from the center, then they must lie on the same circle.
- Practice 3 - Find the point of center to find the midpoint.

#### Math Skill Quizzes

We don't ask for a detailed explanation, you will probably want one though.