2-J

Circles

• circumference and area
• tangents to a circle
• exercises
• answers explained

A circle consists of all the points that are the same distance from one fixed point called the center. That distance is called the radius of the circle. The figure below is a circle of radius 1 unit whose center is at the point O. A, B, C, D, and E, which are each 1 unit from O, are all points on circle O. The word radius is also used to represent any of the line segments joining the center and a point on the circle. The plural of radius is radii. In circle O below, $\overline{OA}$, $\overline{OB}$, $\overline{OC}$, $\overline{OD}$, and $\overline{OE}$ are all radii. If a circle has radius r, each of the radii is r units long. A point is inside a circle if the distance from the center to that point is less than the radius. A point is outside a circle if the distance from the center to that point is greater than the radius.

EXAMPLE 1: In the figure below, O is the center of the circle. To find m∠B and m∠O, observe that since $\overline{OA}$ and $\overline{OB}$ are radii, OA = OB and ΔAOB is isosceles. So, m∠B = 25° and m∠O = 180° – (25° + 25°) = 130°.

Key Fact J1

Any triangle formed by connecting the endpoints of two radii is isosceles.

A chord of a circle is a line segment that has both endpoints on the circle. In the figure at the beginning of this chapter, $\overline{CD}$ and $\overline{BE}$ are chords. A chord such as $\overline{BE}$ that passes through the center is called a diameter. Since BE = EO + OB, a diameter is twice as long as a radius.

Key Fact J2

If d is the diameter and r is the radius of a circle: d = 2r.

Key Fact J3

Diameters are the longest line segments that can be drawn that have both endpoints on or inside a circle.