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. image of a circle with center (h, k) and a point (x, y) on the circle, along with a coordinate plane 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... .

Explanation:

Step1: Determine the lengths of the legs

The horizontal distance between \((h,k)\) and \((x,y)\) is \(|x - h|\) and the vertical distance is \(|y - k|\). So the legs of the right - triangle have lengths \(|x - h|\) and \(|y - k|\).

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem, for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(a^{2}+b^{2}=c^{2}\). Here \(a = |x - h|\), \(b = |y - k|\) and \(c=r\). Substituting these values into the Pythagorean theorem gives \((x - h)^{2}+(y - k)^{2}=r^{2}\)

Answer:

First drop - down: \(|x - h|\) and \(|y - k|\) ; Second drop - down: Pythagorean theorem.