# What is the definition of a coordinate proof? And what is an example?

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Coordinate proof is an algebraic proof of a geometric theorem. In other words, we use numbers (coordinates) instead of points and lines.

In some cases to prove a theorem algebraically, using coordinates, is easier than to come up with logical proof using theorems of geometry.

For example, let's prove using the coordinate method the Midline Theorem that states: Midpoints of sides of any quadrilateral form a parallelogram.

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A coordinate proof is a method of proving geometric theorems using the Cartesian coordinate system. In a coordinate proof, geometric figures and their relationships are represented using algebraic equations and numerical coordinates.

An example of a coordinate proof is proving the midpoint formula for a line segment. Given two points ( A(x_1, y_1) ) and ( B(x_2, y_2) ), the midpoint ( M ) of the line segment connecting ( A ) and ( B ) is ( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ). This can be proven by using the distance formula to show that the distance from ( A ) to ( M ) is equal to the distance from ( B ) to ( M ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A triangle has corners at #(1 ,4 )#, #(7 ,5 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(5 ,-2 )# and a radius of #2 #. Circle B has a center at #(1 ,-4 )# and a radius of #1 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(-4 ,-1 )# and a radius of #3 #. Circle B has a center at #(1 ,3 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?
- A line passes through #(5 ,0 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(1 ,5 )# and #(8 ,7 )#. A second line passes through #(3 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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