From Pythagoras to Circles
Table of Contents
- The Pythagorean Theorem
- Pythagorean Exploration
- Proofs of the Pythagorean Theorem
- The Distance Formula
- Distance between a point and the origin
- The distance formula
- Circles
- Exploring the Circle
- Graphing the Circle
Pythagorean Exploration
Use the area and length tools to explore the relationship between the lengths of side A, B, and C and the squares attached to the triangle.
How is the length of a side related to the area of the square attached to that side?
How are the three squares related to each other?
How can you find the [i]area[/i] of [color=#6aa84f]square C [/color]if you have the [i]area[/i] of [color=#3c78d8]square A[/color] and the [i]area[/i] of [color=#ff0000]square B[/color]?
[left][/left]How can you find the [i]length[/i] of [color=#6aa84f]side C [/color]if you have the [i]length[/i] of [color=#3c78d8]side A[/color] and the [i]length[/i] of [color=#ff0000]side B[/color]?
Write a [i]formula[/i] relating A = [i]length[/i] of [color=#3c78d8]side A[/color], B = [i]l[/i][i]ength[/i] of [color=#ff0000]side B[/color], and C = [i]length[/i] of [color=#6aa84f]side C[/color].
Distance between a point and the origin
Explore the relationship between a point and the distance from the origin![br][br]Use the hints if you need them.
How do the coordinates of the point relate to its distance from the origin?
Write an equation to express the distance (d) between the point and the origin.
Exploring the Circle
Create a circle and connect segments from the center to the edge of the circle. What do you notice about all these segments?
In your own words, define what a circle is (without using equations!)
Draw a circle centered at the origin with a radius 3.
What's an equation that applies to every point [math]P=\left(x,y\right)[/math] on the above circle? [br][br]Hint: think back to [i]finding the distance between a point and the origin.[/i]