Step by step
The question
Question: write the equation of the circle with center (0, 0) and radius 5.
1 Draw the circle
Draw the grid.
Put the center on the grid: (0, 0).
The radius is 5: go 5 from the center.
Every point 5 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 is x − 0.
Go up: from y = 0 to y is y − 0.
Across and up make a right angle.
The radius 5 is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
Put in a, b and c: (x − 0)² + (y − 0)² = 5².
Taking away 0 changes nothing.
So it becomes x² + y² = 5².
5² means 5 times 5.
That is 25.
So the equation is x² + y² = 25.
This works for every circle with center (h, k) and radius r.
4 Check
Let's check with a point on the circle: (5, 0).
Put x = 5 and y = 0 in.
We get 5² + 0².
25 plus 0 is 25, 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.