Chapter 10: Circles

A circle can be defined as a set of points that are equally distant from a single point, known as the center of the circle. A circle's full angle measure is 360 degrees.

A chord is a line that goes from one end of the circle to the other.

A diameter is a chord that travels through the center of a circle. A radius is exactly half the length of a diameter, or from the center to the edge of a circle.

All circles are similar, and circles with the same radius/diameter length are congruent.

Concentric circles are coplanar and share a center. It can be two different sized circles with the same central point.

Two circles can intersect at 0, 1, or 2 points.

The circumference of a circle is 2 x pi x radius, or pi x diameter.

The area of a circle is the radius squared times pi.

A polygon is inscribed in a circle if all of its vertices lie on the circle. The opposite of this is when a circle is CIRCUMSCRIBED about a polygon, as the circle contains all vertices of the polygon.

10.2 Review

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A central angle of a circle is an angle with the vertex being the center of the circle.

An arc is an angle that is a portion of a circle. An arc has two endpoints on the circumference of the circle. There are three types of arcs. A minor arc measures less than 180 degrees. A semicircle is exactly 180 degrees. A major arc is greater than 180 degrees.

The Arc Addition Postulate states that an arc formed by two adjacent arcs is equal to the sum of those two arcs.

TO FIND THE LENGTH OF A CENTRAL ARC IN UNITS, use the following formula: Arc = pi(diameter)(x)/360, where x is the angle measure of the central arc.

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As we know, a chord that is not a diameter divides a circle into a minor and major arc.

In the same or another congruent circle, two minor arcs are congruent if their corresponding chords are congruent.

If a diameter or radius is perpendicular to a chord, then it bisects that chord and arc into two equal pieces. The opposite of this is also true; a perpendicular bisector of a chord is a diameter or radius of that circle.

Two chords are also congruent if they are equidistant from the center of the same (or a congruent) circle.

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Inscribed angles are angles that have all three points on the circumference of the circle. The center of the circle can be either inside, on, or outside the inscribed angle.

THE INSCIRBED ANGLE THEOREM STATES THAT an inscribed angle's arc is exactly double the measure of the angle. For example, if inscribed angle ABC creates arc AC, and that arc has a measure of 50 degrees, then angle ABC has a measure of 25 degrees.

If two inscribed angles INTERCEPT the same arc (that means they create that arc), then those two angles are equal(this is common sense).

If an inscribed angle creates an arc that is a semicircle, then the angle opposite the diameter is a right angle.

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

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A tangent is a line that intersects a circle in exactly one point, which is called the point of tangency. If a line is tangent to two different circles, then that line is called a common tangent.

A line is only tangent to a circle if it is perpendicular to the radius of the circle.

If two line segments from the same exterior point are tangent to a circle, then those line segments are congruent.

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Secants are lines that intersect a circle at exactly 2 points.

Intersection of Secants: If 2 secants intersect in the interior of a circle then the measure of the angle formed is average of the measure of arcs intercepted by the angle and its vertical angle.

Intersection of Secants & Tangents: When a tangent and a secant intersect at the point of tangency then the measure of each angle formed is half the measure of its intercepted arc.

Intersections on exterior of circle: When a tangent and a secant intersect in the exterior of a circle then measure of angle formed is half the measure of the difference of the measures of its intercepted arcs.

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When chords intersect inside a circle, they create chord segments. Two chords will create four segments. When you multiply the lengths of two chord segments, the products for both chords will be equal (see slide 56).

A secant segment is a segments of a secant that has one endpoint on the circle. When such a segment is on the outside of a circle, it is called an external secant segment.

When two secants intersect outside the circle, new properties arise. The products of the external secant segment and the entire secant will be the same for both lines (see slide 59).

If a tangent and a secant intersect outside the circle, then the product of the length of the external secant segment and the entire secant is equal to the square of the length of the tangent.

10.8 Review

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

When given a center and radius, you can find the equation of the circle by directly inputting the values given into the equation written above.

When given a center and a point you can find the equation by using the distance formula. That will provide you with the radius. Once you find the radius you can input all the values in the above equation and solve.

Once again we provide you with the standard form for the equation of a circle.

r^2 = (x - h)^2 + (y - k)^2 , where r = radius. This form of writing an equation is known as center-radius form, or standard form. REMEMBER THAT THE SIGNS ARE NEGATIVE H AND NEGATIVE K, SO THE SIGNS FOR THE COORDINATES OF THE CENTER WILL BE FLIPPED.