Cirlces Page 4a (Pythagoras)

Page 4a

Derivation by Pythagorean Theorem

Let's see how the Pythagorean Theorem can be used to derive the equation of a circle centred at the origin. Figure 1
[CIRLCE]

The circle in Figure 1 is centred at the origin with radius r. Let (x, y) be any point on the circle.

Notice in the action figure below that within this circle you can draw a right triangle with r as the hypotenuse and x and y as the lengths of the legs.

Action Figure 1
Using the action figure on the left, click and hold on the small blue square and move it around the circle. Notice the right triangle can be drawn for every point (x, y) on the circle.

Now that we have created a right triangle, we can use the Pythagorean Theorem to find an equation of a circle with a centre at the origin. This equation is: x² + y² = r².

Notice that this equation is the same as the standard equation for a circle centred at the origin found using the Distance Formula.

This is because the Distance Formula and the Pythagorean Theorem are equivalent. That is, you can derive either one easily from the other.

Let's see how the Pythagorean Theorem can be used to derive the equation of a circle centred at the origin.	Figure 1
The circle in Figure 1 is centred at the origin with radius r. Let (x, y) be any point on the circle.
Notice in the action figure below that within this circle you can draw a right triangle with r as the hypotenuse and x and y as the lengths of the legs.

Action Figure 1	Using the action figure on the left, click and hold on the small blue square and move it around the circle. Notice the right triangle can be drawn for every point (x, y) on the circle.
	Now that we have created a right triangle, we can use the Pythagorean Theorem to find an equation of a circle with a centre at the origin. This equation is: x² + y² = r².
Notice that this equation is the same as the standard equation for a circle centred at the origin found using the Distance Formula. This is because the Distance Formula and the Pythagorean Theorem are equivalent. That is, you can derive either one easily from the other.