Question

move the answers to the boxes to complete the sentences. the pythagorean theorem is a² + b² = c². use any point on the circle (x, y) to create a right - triangle that has a hypotenuse with a

Explanation:

Step1: Identify right - triangle sides

Let the center of the circle be at the origin \((0,0)\) (assumed from the context of using the Pythagorean theorem). If we consider a point \((x,y)\) on the circle and the right - triangle formed with the radius as the hypotenuse. Let the horizontal distance from the center to the point be \(a\) and the vertical distance be \(b\).

Step2: Apply Pythagorean theorem

If the radius of the circle is \(r\), and for a right - triangle with sides \(a\) and \(b\) and hypotenuse \(r\), by the Pythagorean theorem \(a^{2}+b^{2}=r^{2}\). For example, if the radius \(r = 5\) and we take a point \((x,y)\) on the circle, and assume the horizontal displacement \(a=x\) and vertical displacement \(b = y\), then \(x^{2}+y^{2}=25\).

Answer:

The relationship between the coordinates \((x,y)\) of a point on the circle and the radius \(r\) of the circle is \(x^{2}+y^{2}=r^{2}\) based on the Pythagorean theorem.