Question

zain draws a circle with radius r and center (h,k) in the coordinate plane. he places the point (x,y) on the circle. how can zain use his drawing to derive the general equation of a circle in standard form? use the drop - down menus to explain your answer. click the arrows to choose an answer from each menu. using any center point (h,k) and any point on the circle (x,y), zain can draw a right triangle that has a hypotenuse of length r and legs of lengths choose. then, zain can derive the general equation of a circle in standard form by applying the choose. pythagorean theorem, triangle sum theorem, area formula

Explanation:

Step1: Recall distance concept

The distance between the center $(h,k)$ and a point $(x,y)$ on the circle is the radius $r$. We can consider the right - triangle formed. The horizontal distance between the points is $|x - h|$ and the vertical distance is $|y - k|$.

Step2: Apply Pythagorean theorem

In a right - triangle with legs of lengths $a=|x - h|$ and $b = |y - k|$ and hypotenuse of length $r$, by the Pythagorean theorem $a^{2}+b^{2}=r^{2}$. Substituting $a=x - h$ and $b=y - k$, we get $(x - h)^{2}+(y - k)^{2}=r^{2}$.

Answer:

The legs of the right - triangle have lengths $|x - h|$ and $|y - k|$ and Zain can derive the general equation of a circle in standard form by applying the Pythagorean theorem. The standard form of the equation of the circle is $(x - h)^{2}+(y - k)^{2}=r^{2}$.