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 right - triangle sides

The horizontal distance between the center \((h,k)\) and the point \((x,y)\) on the circle is \(|x - h|\) and the vertical distance is \(|y - k|\). These are the lengths of the legs of the right - triangle.

Step2: Apply theorem

By the Pythagorean theorem, for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), we have \(a^{2}+b^{2}=c^{2}\). Here, \(a = x - h\), \(b=y - k\) and \(c = r\), so \((x - h)^{2}+(y - k)^{2}=r^{2}\).

Answer:

For the first blank: \(|x - h|\) and \(|y - k|\); For the second blank: Pythagorean theorem.