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

How long is the hypotenuse of the right triangle in the image?

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)

Fill in the Blanks

Find the distance between (2, 3) and (7, 16).

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.

Fill in the Blanks

What is the midpoint of (0, 8) and (2, -4)?

​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

Fill in the Blanks

What is the volume of a rectangular box that is 3 feet long, 2 feet wide, and 6 inches tall?