# Coordinate Proof Worksheets

When a proof centralizes it argument around the use of a geometric theorem that uses generalized points on a plane, we call this a coordinate proof. The general principles are that these types of proofs rely on the use of numbers in the form of coordinates and not on lines or points. When presenting coordinate proofs the common tactic that used is assigning a variable to a series of coordinates and using that point as a reference for use within the distance formula or midpoint formula. Often proofs are much easier when solved algebraically, but in some cases when the exact details of the location of points or line position is ambiguous these come in very handy. Students can use these lessons and worksheets to learn how to write coordinate proofs in a wide variety of situations.

### Aligned Standard: High School Geometry - HSG-GPE.B.4

- 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?

- Answer Keys - These are for all the unlocked materials above.

### 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.

- Quiz 1 - State: Yes or No. Does the point (7, -25) lie on the circle centered at (-3, -6), and containing the point (-8, 18)?
- Quiz 2 - This has powerful applications for some forms of trig.
- Quiz 3 - We playing Battleship with circles here.

### 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 formula 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. This 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 formula, you can prove all types of theorems.

As a general guide when we are attempting to prove that a geometric shape has congruent sides, we would most often use the distance formula or Pythagorean Theorem, if it is a triangle. They both help us determine the exact length of a side. When you are looking to prove that the sides are parallel you can do so by looking at their slope. If both lines have the same slope, they will never run into each other and a parallel. If you are looking to prove that shape has a right angle, just look for perpendicular lines. If you are not given enough information to determine that two lines are perpendicular, you can prove it by determining their slope. Two lines are perpendicular if the slopes of their lines are opposite reciprocals.