Cirlces Page 5a (Pythagoras)

Page 5a

Derivation of Standard Form by the Pythagorean Theorem

The Pythagorean Theorem can also be used to find the equations of circles centred at any point, (h, k). Figure 1
[SIMPLE

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

Answer the following questions about the picture in Figure 1 to discover lengths of legs a and b in the right triangle.

You will then use this information to arrive at the standard form of an equation of a circle.

Now we can label the dimensions of the right triangle as shown in Figure 2: Figure 2
[TRIANGLE in CIRCLE]

Using the Pythagorean Theorem, a² + b² = c², we can derive the following equation:(x - h)² + (y - k)² = r²

Notice that this is the same standard equation for a circle that we found using the distance formula. Again we see that the Distance Formula and the Pythagorean Theorem are equivalent.

The Pythagorean Theorem can also be used to find the equations of circles centred at any point, (h, k).	Figure 1
The circle in Figure 1 is centred at (h, k) with radius r. Let (x, y) be any point on the circle.
Answer the following questions about the picture in Figure 1 to discover lengths of legs a and b in the right triangle.
You will then use this information to arrive at the standard form of an equation of a circle.

Now we can label the dimensions of the right triangle as shown in Figure 2:	Figure 2
Using the Pythagorean Theorem, a² + b² = c², we can derive the following equation:(x - h)² + (y - k)² = r²
Notice that this is the same standard equation for a circle that we found using the distance formula. Again we see that the Distance Formula and the Pythagorean Theorem are equivalent.