# Pythagorean Theorem Examples

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More Lessons for High School Geometry
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A series of free, online High School Geometry Video Lessons.

Videos, worksheets, and activities to help Geometry students.

In these lessons, we will learn

- how to derive the distance formula using the Pythagorean Theorem
- how to derive the equation of a unit circle from the Pythagorean Theorem
- how to derive and memorize the coordinates of the unit circle

### Distance Formula

Using what we know about the Pythagorean theorem, we are able to derive the distance formula which is used to find the straight distance between two points in a coordinate plane. The distance formula is a standard formula that allows us to plug a set of coordinates into the formula and easily calculate the distance between the two.

This video show how to use the distance formula to determine the distance between two points. It also shows how it is derived from the Pythagorean theorem.

How to derive the equation for the distance between two points using the Pythagorean Theorem.

### Equation of a Circle

A circle can't be represented by a function, as proved by the vertical line test. However, we can obtain an equation that describes the full circle by using the distance formula between the given center coordinates and any point along the circumference of the circle. Once we have derived this equation of a circle, we can apply it to any other circle you may come across in a coordinate plane.

This lesson shows two examples regarding how to find the equation of a circle centered at the origin.

How to derive the equation for a circle using the distance formula.

The equation of a circle. This equation comes from the Pythagorean Theorem.

Center: (h, k)

Radius: r

Circle Equation: (x-h)

^{2} + (y-k)

^{2} = r

^{2}
Examples of coming up with the equation of a circle and graphing a given equation of a circle.

### Calculating Coordinates in the Unit Circle

The unit circle is a circle that has a radius of one and is centered at the origin of the coordinate plane. It is a concept that frequently occurs in many of the math subjects, especially those where Trigonometry is used. Questions asking about unit circle coordinates often give an unknown coordinate and require us to use the properties of a unit circle to calculate these coordinates.

A way to remember the Entire Unit Circle for Trigonometry.

Deriving Values on the Unit Circle.

This video shows how to derive the values in the first quadrant of the unit circle by using geometry and the Pythagorean theorem.