Question

the midpoint of the diameter of a circle gives the center of the circle. when solving for the radius, you need to the distance from the center to any point on the circle. a circles equation in standard form is derived from the formula. the of a circle is the longest chord and it passes through the center. the distance formula is used to find the between the center and any point on the circle.

Explanation:

Step1: Recall circle - center property

The mid - point of the diameter of a circle is the center of the circle.

Step2: Recall radius definition

The radius is the distance from the center to any point on the circle, so when solving for the radius, you measure this distance.

Step3: Recall circle equation derivation

The standard form of a circle's equation \((x - h)^2+(y - k)^2=r^2\) is derived from the distance formula, where \((h,k)\) is the center and \((x,y)\) is a point on the circle, and \(r\) is the radius.

Step4: Recall diameter property

The diameter of a circle is the longest chord and it passes through the center.

Step5: Recall distance formula use

The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) is used to find the distance between the center \((h,k)\) (i.e., \((x_1,y_1)\)) and any point \((x,y)\) (i.e., \((x_2,y_2)\)) on the circle.

Answer:

1. center
2. measure
3. distance
4. diameter
5. distance