Question

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), point (x,y), radius r, coordinate axes 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 leg lengths

The horizontal leg length is the difference in x - coordinates: \(|x - h|\), and the vertical leg length is the difference in y - coordinates: \(|y - k|\). So the legs have lengths \(|x - h|\) and \(|y - k|\) (or \((x - h)\) and \((y - k)\) when squared, as absolute value squared is the same as the square of the difference).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here, \(a = |x - h|\), \(b = |y - k|\), and \(c = r\). Squaring both sides (since \(|a|^{2}=a^{2}\)), we get \((x - h)^{2}+(y - k)^{2}=r^{2}\), which is the standard form of the equation of a circle. So we apply the Pythagorean theorem.

Answer:

First drop - down: \(|x - h|\) and \(|y - k|\) (or \((x - h)\) and \((y - k)\))
Second drop - down: Pythagorean theorem
The standard form of the circle's equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\)