Sure, let's break down how Zain can derive the equation of a circle step-by-step:
1. Draw a right triangle:
Using any center point [tex]\((h, k)\)[/tex] and any point on the circle [tex]\((x, y)\)[/tex], Zain can draw a right triangle that has a hypotenuse of length [tex]\(r\)[/tex] and legs of lengths [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex].
2. Apply the Pythagorean Theorem:
In this right triangle, the hypotenuse is the distance from the center of the circle to the point on the circle, which is the radius [tex]\(r\)[/tex]. According to the Pythagorean Theorem, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
So, we have:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
3. Conclusion:
The equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] is the general equation of a circle in standard form.
Putting these steps together:
1. Using any center point [tex]\((h, k)\)[/tex] and any point on the circle [tex]\((x, y)\)[/tex], Zain can draw a right triangle that has a hypotenuse of length [tex]\(r\)[/tex] and legs of lengths [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex].
2. Then, Zain can derive the general equation of a circle in standard form by applying the Pythagorean Theorem.
Thus, the general equation of the circle in standard form that Zain will derive is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
