Step by step
The question
Question: write the equation of the circle with center (0, 0) and radius 7.
1 Draw the circle
Draw the grid.
Put the center on the grid: (0, 0).
The radius is 7: go 7 from the center.
Every point 7 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across, from x = 0 to x.
How far?
The end take away the start: x − 0.
Go up, from y = 0 to y.
The end take away the start: y − 0.
Across and up make a right angle.
The radius 7, from the center to (x, y), is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
Put in a, b and c: a is the across, b is the up, c is the radius 7.
(x − 0)² + (y − 0)² = 7².
Taking away 0 changes nothing.
So it becomes x² + y² = 7².
7² means 7 times 7.
That is 49.
So the equation is x² + y² = 49.
This works for every circle.
Write h for the x of the center, k for its y, and r for the radius.
4 Check
Let's check with a point on the circle: (7, 0), 7 to the right of the center.
Put x = 7 and y = 0 in.
We get 7² + 0².
49 plus 0 is 49, the number on the right.
Right!
Let's look back.
First: draw the circle.
Then: across and up.
Then: use a² + b² = c².
Last: check.