Exploring Circles

Definitions

Notation: if the center is P then the circle can be denoted by סּP The points inside the circle form the circle's interior. The point outside the circle form its exterior. ● A circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. NOTE: Each point on the circle is equidistant from the center.

A chord of a circle is a segment whose endpoints are on the circle. ● ● A radius of a circle is a segment whose endpoints are the center of the circle and another point on the circle. ● ● ● ● ● A diameter of a circle is a chord that passes through the center.

Remarks: 1By the definition of a circle, all radii (plural of radius) of a circle are congruent. 3 A diameter is always a chord, BUT a chord is NOT always a diameter. 2 There are two (2) definitions for radius: it is the segment; and the length of the segment

● ● ● ● A tangent is a line that Intersects the circle at exactly one point. T PR S This point is called the point of tangency. Q Ex. is a tangent of A secant is a line that intersects the circle at two points. Ex. is a secant of

● P Concentric circles are distinct coplanar circles with the same center. Congruent circles are distinct circles with equal radii.

If every vertex of a polygon is a point on a circle, then the polygon is INSCRIBED in the circle… … and the circle is CIRCUMSCRIBED ABOUT the polygon.

● P ● Q ● Internally tangent circles are tangent circles on the same side of the tangent line. ● P ● Q ● Externally tangent circles are tangent circles on opposite sides of the tangent line.

● ● ● ● B PR S Q ● ● A An arc of a circle is composed of any two points on the circle and all the points between them. Ex. A minor arc of a circle is an arc that is less than a semicircle. A semicircle is an arc whose endpoints are the endpoints of a diameter. ● ● ● ● B PR S Q ●● A ● ● ● ● B P R S Q ● ● A Note: and are the same major arc. A major arc of a circle is an arc that is more than a semicircle.

● ● ● ● B PR S Q ● ● A An inscribed angle is an angle whose vertex is point on the circle and whose sides are chords. A central angle is an angle whose vertex is the center of the circle and whose sides are radii.

A C D E F P T Exercises:

Arc Measure

Arc Measures A full circle has an arc measure of 360 o. C B A The arc measure of a minor or major arc is the same as the measure of the central angle. semicircle ACB major arc ABC minor arc AC 45 o 315 o A semicircle has an arc measure of 180 o.

Exercises

Exercises:

Always, Sometimes, Never 1. A chord is a diameter.S 2. A radius is a chord.N 3. A diameter is contained in a secant.A 4. A tangent contains a chord.N 5. Concentric circles are congruent.N 6. Congruent circles have equal radii.A

Problem Solving 1.Find x. 2. Radius of circle D is 6cm. Radius of circle E is 8cm. Find DE. 3.In a circle, two radii form a 60  arc. If the chord determined by the radii has a length of 6, find the diameter of the circle.

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