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9.8: Chapter 9 Review

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Keywords & Theorems

Circle
The set of all points that are the same distance away from a specific point
Center
The set of all points that are the same distance away from a specific point, called the center.
Radius
The distance from the center to the circle.
Chord
A line segment whose endpoints are on a circle.
Diameter
A chord that passes through the center of the circle.
Secant
A line that intersects a circle in two points.
Tangent
A line that intersects a circle in exactly one point.
Point of Tangency
The point where the tangent line touches the circle.
Congruent Circles
Two circles with the same radius, but different centers.
Concentric Circles
When two circles have the same center, but different radii.
Tangent to a Circle Theorem
A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
Theorem 9-2
If two tangent segments are drawn from the same external point, then the segments are equal.
Central Angle
The angle formed by two radii of the circle with its vertex at the center of the circle.
Arc
A section of the circle.
Semicircle
An arc that measures 180^\circ.
Minor Arc
An arc that is less than 180^\circ.
Major Arc
An arc that is greater than 180^\circ. Always use 3 letters to label a major arc.
Congruent Arcs
Two arcs are congruent if their central angles are congruent.
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two
Theorem 9-3
In the same circle or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.
Theorem 9-4
The perpendicular bisector of a chord is also a diameter.
Theorem 9-5
If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.
Theorem 9-6
In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.
Inscribed Angle
An angle with its vertex is the circle and its sides contain chords.
Intercepted Arc
The arc that is on the interior of the inscribed angle and whose endpoints are on the angle.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
Theorem 9-8
Inscribed angles that intercept the same arc are congruent.
Theorem 9-9
An angle that intercepts a semicircle is a right angle.
Inscribed Polygon
A polygon where every vertex is on a circle.
Theorem 9-10
A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
Theorem 9-11
The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.
Theorem 9-12
The measure of the angle formed by two chords that intersect inside a circle is the average of the measure of the intercepted arcs.
Theorem 9-13
The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.
Theorem 9-14
The product of the segments of one chord is equal to the product of segments of the second chord.
Theorem 9-15
If two secants are drawn from a common point outside a circle and the segments are labeled as above, then a(a+b)=c(c+d).
Theorem 9-16
If a tangent and a secant are drawn from a common point outside the circle (and the segments are labeled like the picture to the left), then a^2=b(b+c).
Standard Equation of a Circle
The standard equation of a circle with center (h, k) and radius r is r^2=(x-h)^2+(y-k)^2.

Vocabulary

Match the description with the correct label.

  1. minor arc - A. \overline{CD}
  2. chord - B. \overline{AD}
  3. tangent line - C. \overleftrightarrow{CB}
  4. central angle - D. \overleftrightarrow{EF}
  5. secant - E. A
  6. radius - F. D
  7. inscribed angle - G. \angle BAD
  8. center - H. \angle BCD
  9. major arc - I. \widehat{BD}
  10. point of tangency - J. \widehat{BCD}

Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9694.

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