Geometry Chapter 12: Circles



Geometry Chapter 12: Circles Name: ________________________________

12.1 Tangent Lines

Objectives: To use the relationship between a radius and a tangent. To use the relationship between two tangents from one point

Tangent to a Circle:

Point of Tangency:

Theorem 12-1:

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

|1. [pic]is tangent to [pic]at point A. Find the value of x. |2. [pic]is tangent to[pic]. Find the value of x. |

|[pic] |[pic] |

|3. Find the value of x |4. Find the value of x |

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|5. Find the value of x |

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|6. [pic]is inscribed in quadrilateral XYZW. Find the perimeter of XYZW. |7. [pic]is inscribed in [pic] [pic]has a perimeter of 88 cm. Find QY. |

|[pic] |[pic] |

|8. [pic]has radius 5. Point P is outside[pic]such that PO = 12, and point A |9. If OA = 4, AP =7, and OP = 8, is [pic]tangent to [pic]at A? |

|is on [pic]such that PA = 12. Is [pic]tangent to [pic] at A? Explain. |[pic] |

|[pic] | |

|10. A belt fits tightly around two circular pulleys, as shown. Find the |11. In the diagram, ZY is tangent to circles O and P. Find the value of x. |

|distance between the centers of the pulleys. Round your answer to the nearest |[pic] |

|tenth. | |

|[pic] | |

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12.2 Chords and Arcs

Objectives: To use congruent chords, arcs, and central angles. To recognize properties of lines through the center of a circle

|Within a circle, or in congruent circles… |

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|And |

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|In a circle, a diameter that is perpendicular to a chord bisects the chord and|The perpendicular bisector of a chord contains the center of the circle. |

|its arcs. | |

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|The converse is also true. | |

|1. Find AB. |2. Find the value of x. |3a. Find the length of the chord. |

|[pic] |[pic] |[pic] |

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| | |3b. Find the distance from the midpoint of the chord to the midpoint of its |

| | |minor arc. |

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|5. |6. [pic] |

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|7. Radius [pic]bisects [pic] What can you conclude, and why? |8. P and Q are points on [pic]. The distance from O to [pic]is 15 in., and PQ|

|[pic] |= 16 in. Find the radius of [pic]. |

| |[pic] |

|₪ [pic] by the | |

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|₪ [pic] because | |

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|₪ [pic] because | |

12.3 Inscribed Angles

Objectives: To find the measure of an inscribed angle. To find the measure of an angle formed by a tangent and a chord

|If the vertex of an angle is on the circle, and the sides of the angle are |The measure of an inscribed angle is half the measure of its intercepted arc. |

|chords of the circle, then the angle is an inscribed angle. | |

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|The arc cut off by the angle is called the intercepted arc. | |

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|There are three corollaries to the theorem. |The measure of an angle formed by a tangent and a chord is half the measure of|

| |the intercepted arc. |

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|1. Find x. |[pic] |

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|2. Find y. | |

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|3. Find [pic] | |

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|4. Find x |5. Find x |

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|6. Find x |7. Find x |

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|8. Find the values of a and b. |9. Find the measure of each numbered angle. |

|[pic] |[pic] |

|10. [pic]and [pic]are diameters of circle A. [pic]is tangent to [pic]at point R. Find[pic] |

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|11. Find x |12. Find x, y, and z |

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12.4 Angle Measures and Segment Lengths

Objectives: To find the measures of angles formed by chords, secants, and tangents. To find the lengths of segments associated with circles.

|A secant is a line that intersects a circle at two |The measure of an angle formed by two lines that |The measure of an angle formed by two lines that |

|points. |intersect a circle is half the sum of the measures |intersect outside a circle is half the difference of|

| |of the intercepted arcs: |the measures of the intercepted arcs: |

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|For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and |

|circle. |

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|1. Find the value of W. |2. Find the value of x to the nearest |3. Find the value of x. |4. Find the value of x. |

|[pic] |tenth. |[pic] |[pic] |

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|5. Find the value of x. |6. Find the value of x. |7. Find the value of x. |

|[pic] |[pic] |[pic] |

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|8. An advertising agency wants a frontal photo of a “flying saucer” ride at an|9. A tram travels from point A to point B along the arc of a circle with a |

|amusement park. The photographer stands at the vertex of the angle formed by |radius of 125 ft. Find the shortest distance from point A to point B. |

|tangents to the “flying saucer.” What is the measure of the arc that will be | |

|in the photograph? In the diagram, the photographer stands at point T. | |

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12.5 Circles in the Coordinate Plane

Objectives: To write an equation of a circle. To find the center and radius of a circle on a coordinate plane

|The equation of a circle with center[pic]and radius r is |

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|This is called the standard form of an equation of a circle. |

|1. Write the standard equation of a circle with center [pic]and radius |2. A diagram locates a radio tower at [pic]on a coordinate grid where each unit |

|[pic] |represents 1 mile. The radio signal’s range is 80 miles. Find an equation that |

| |describes the position and range of the tower. |

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|3. Write the standard equation of a circle with center [pic]that passes |4. Write the standard equation of a circle with center [pic]that passes through the|

|through the point[pic]. |point[pic]. |

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|5. Find the center and radius of the circle with equation [pic] Then |6. Find the center and radius of the circle with equation [pic] Then graph the |

|graph the circle. |circle. |

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|[pic] |[pic] |

|7. When you make a call on a cell phone, a tower receives the call. In the diagram, the centers of circles O, A, and B are locations of cellular towers. |

|Write an equation that describes the position and range of tower A. |

|[pic] |

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