Question

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 legs

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 lengths of the legs are \(|x - h|\) and \(|y - k|\) (or \((x - h)\) and \((y - k)\) when considering the squared values for the Pythagorean theorem).

Step2: Identify the theorem for circle equation

To derive the standard form of the circle's equation \((x - h)^2+(y - k)^2 = r^2\), we use 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 \((x - h)^{2}+(y - k)^{2}=r^{2}\).

Answer:

1. For the first "Choose...": \(|x - h|\) and \(|y - k|\) (or \((x - h)\) and \((y - k)\))
2. For the second "Choose...": Pythagorean theorem