Rectangular Coordinates, Geometric Formulas, and Graphs

Rectangular Coordinates

How do we work with a Cartesian Plane?

The Parts of a Cartesian Plane

• The x-axis is the horizontal axis (remember that "horizontal" comes from the word "horizon" like what you see when you look outside).

• The y-axis is the vertical axis (straight up and down).

• The origin is the point (0, 0)

Quadrants of a Cartesian Plane

The Cartesian Plane has quadrants 1 - 4 which proceed subsequently in a C-shape.

Pythagorean Theorem

For all right triangles, the lengths of the squares of the legs equals the length of the hypotenuse squared.

Pythagorean Practice

Find x. The triangle is a right triangle.

Fill in the Blanks

Distance Formula

The distance formula can be used to find the distance between two points or the length of a segment using its endpoints.

Practice Problems

• Find the distance between (1, 9) and (-5, 2)

• Find the distance between (2, 4) and (8, 3)

• Find the distance between (-8, 1) and (10, 5)

Verifying Right Triangles on Cartesian Planes

• Use the distance formula to find the lengths of the sides of the triangle.

• Plug those distances into the Pythagorean Theorem.

• If the resulting statement is true, then it is a right triangle.

​Midpoint Formula

The Midpoint Formula can be used to find the coordinates of the point that is exactly in the middle of two other points.​

​Translating Points

• ​The translation to the right can be written

​

​ So, what is the translation of B to

B'?

​

​ The x movement is based on addition or subtraction to the x value and y movement is based on addition or subtraction to the y value.

​When Using a Geometric Formula...

• Use the CORRECT formula​

  Plug the numbers in to their respective spots using parentheses

• ​Then, solve for the missing value