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) and a point (x,y) on it, with 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: Analyze the right triangle sides

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 considering the distance formula, as squaring will eliminate the absolute value).

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 \(c = r\). So \((x - h)^{2}+(y - k)^{2}=r^{2}\) (since \((|x - h|)^{2}=(x - h)^{2}\) and \((|y - k|)^{2}=(y - k)^{2}\)).

For the first drop - down (legs of lengths): The horizontal leg is the difference in the x - coordinates of the center \((h,k)\) and the point on the circle \((x,y)\), so \(|x - h|\) (or \(x - h\)) and the vertical leg is the difference in the y - coordinates, so \(|y - k|\) (or \(y - k\)). So the legs have lengths \(|x - h|\) and \(|y - k|\) (or \(x - h\) and \(y - k\)).

For the second drop - down (applying the): The Pythagorean theorem, which relates the lengths of the legs and the hypotenuse of a right triangle.

Answer:

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