Let’s derive the equation step-by-step. Here is the step-by-step solution:
1. Identify Segment [tex]\( AB \)[/tex]: In the context of a circle, segment [tex]\( AB \)[/tex] is the radius [tex]\( r \)[/tex] of the circle. The circle is centered at [tex]\( A(h, k) \)[/tex].
2. Determine the Length of Leg [tex]\( AC \)[/tex]: For a point [tex]\( B(x, y) \)[/tex] on the circle, the horizontal distance from the center [tex]\( A(h, k) \)[/tex] to the point [tex]\( B(x, y) \)[/tex] is [tex]\( |x - h| \)[/tex].
3. Determine the Length of Leg [tex]\( BC \)[/tex]: Similarly, the vertical distance from the center [tex]\( A(h, k) \)[/tex] to the point [tex]\( B(x, y) \)[/tex] is [tex]\( |y - k| \)[/tex].
4. Apply the Pythagorean Theorem: Since [tex]\( \triangle ABC \)[/tex] is a right triangle with legs [tex]\( AC \)[/tex] and [tex]\( BC \)[/tex], and hypotenuse [tex]\( AB \)[/tex], we can use the Pythagorean theorem:
[tex]\[
(AC)^2 + (BC)^2 = (AB)^2
\][/tex]
5. Substitute the Lengths into the Pythagorean Theorem:
- [tex]\( AC = |x - h| \)[/tex], so [tex]\( (AC)^2 = (x - h)^2 \)[/tex]
- [tex]\( BC = |y - k| \)[/tex], so [tex]\( (BC)^2 = (y - k)^2 \)[/tex]
- [tex]\( AB = r \)[/tex], the radius of the circle, so [tex]\( (AB)^2 = r^2 \)[/tex]
Therefore, the equation becomes:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
6. Standard Form of the Equation of a Circle: The derived equation
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
is the standard form of the equation of a circle centered at [tex]\( A(h, k) \)[/tex] with radius [tex]\( r \)[/tex].
