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Coordinate Proofs

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Coordinate Proofs

Study Guide

Key Definition

Coordinate proofs use "generalized" points on the Cartesian Plane to establish geometric theorems. These proofs often employ $x$ and $y$ coordinates of points, and utilize the midpoint or distance formulas.

Important Notes

Coordinate proofs can be used to prove the Triangle Midsegment Theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half the length.
The method usually involves assigning variables to the coordinates of one or more points.
The slope is calculated to determine parallelism.

Mathematical Notation

$(a, b)$ is used to denote a point in the Cartesian plane, where $a$ is the $x$-coordinate and $b$ is the $y$-coordinate.

$\overline{XY}$ represents the line segment from point $X$ to point $Y$.

Remember to use proper notation when solving problems.

Why It Works

The reason coordinate proofs work is due to the properties of the Cartesian plane and the formulas we use for distance and midpoint. The plane's grid structure allows us to measure geometric properties like length and distance, while the formulas give us precise computations.

Remember

While proving geometric theorems, always remember to clearly identify your points, use appropriate formulas, and follow the logical flow of the proof.

Quick Reference

Midpoint Formula:$\left( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} \right)$

Distance Formula:$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Understanding Coordinate Proofs

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Video explanation of this concept

Beginner

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Beginner Explanation

Coordinate proofs involve the use of coordinates and formulas to establish geometric theorems. They are particularly useful in proving theorems related to triangles and lines.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the coordinates of a point are $(a, b)$, what will be the coordinates of its midpoint with the origin?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the coordinates of the vertices of a triangle are $(0, 0)$, $(a, 0)$ and $(b, c)$, what are the coordinates of the midpoints of its sides?

Challenge Quiz

Single Choice Quiz

Advanced

If the coordinates of the vertices of a triangle are $(0, 0)$, $(a, 0)$ and $(b, c)$, is the segment connecting the midpoints of the sides parallel to the $x$-axis?

Recap

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