Circles and Ellipses

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Derive Equation of a Circle from Pythagorean Theorem

Lesson

Recall that in any right triangle, the sum of the squares of the two sides is equal to the square of the hypotenuse. This relationship is proven in the Pythagorean Theorem.

For example, in this right triangle to the left $a^2+b^2=c^2$`a`2+`b`2=`c`2.

In the coordinate plane, the distance between two points can be found by using the Pythagorean Theorem.

Recall also that a circle is the set of all points equidistant from a single point.

Use the facts above as well as the applet below to answer the questions that follow. Your answers should help you derive the equation for a circle in the coordinate plane.

Note: The point $(x,y)$(`x`,`y`) represents any point on the circle, and the point $(h,k)$(`h`,`k`) represents the center of the circle with radius $r$`r`, where $r>0$`r`>0.

1. Move the slider that says "Slide Me!". What shape is formed?

2. Explain what is meant by $|x-h|$|`x`−`h`| and $|y-k|$|`y`−`k`|.

3. Write an equation that relates $|x-h|$|`x`−`h`|, $|y-k|$|`y`−`k`|, and $r$`r`.

4. Does this relationship change as the black point is moved around the circle? Explain your reasoning.

Apply the geometry of conic sections

Apply the geometry of conic sections in solving problems