PA and PB are tangent to C. Use the figure for Exercises 1–3.

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Presentation transcript:

PA and PB are tangent to C. Use the figure for Exercises 1–3. Tangent Lines Lesson 12-1 Lesson Quiz . PA and PB are tangent to C. Use the figure for Exercises 1–3. 1. Find the value of x. 2. Find the perimeter of quadrilateral PACB. 3. Find CP. HJ is tangent to A and to B. Use the figure for Exercises 4 and 5. 4. Find AB to the nearest tenth. 5. What type of special quadrilateral is AHJB? Explain how you know. 87.2 82 cm 29 cm . . 20.4 cm Trapezoid; the tangent line forms right angles at vertices H and J, so HA || JB. Because HA = JB, AHJB is not a parallelogram but a trapezoid. / 12-2

Find the value of each variable. Leave your answer in simplest Chords and Arcs Lesson 12-2 Check Skills You’ll Need (For help, go to Lesson 8-2.) Find the value of each variable. Leave your answer in simplest radical form. 1. 2. 3. Check Skills You’ll Need 12-2

A segment whose endpoints are on a circle is called a chord. Chords and Arcs Lesson 12-2 Notes A segment whose endpoints are on a circle is called a chord. 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

Chords and Arcs Lesson 12-2 Notes 12-2

In the diagram, radius OX bisects AOB. What can you conclude? Chords and Arcs Lesson 12-2 Additional Examples Using Theorem 12-4 In the diagram, radius OX bisects AOB. What can you conclude? AOX BOX by the definition of an angle bisector. AX BX because congruent central angles have congruent chords. AX BX because congruent chords have congruent arcs. Quick Check 12-2

QS = QR + RS Segment Addition Postulate Chords and Arcs Lesson 12-2 Additional Examples Using Theorem 12-5 Find AB. QS = QR + RS Segment Addition Postulate QS = 7 + 7 Substitute. QS = 14 Simplify. AB = QS Chords that are equidistant from the center of a circle are congruent. AB = 14 Substitute 14 for QS. Quick Check 12-2

Using Diameters and Chords Chords and Arcs Lesson 12-2 Additional Examples Using Diameters and Chords P and Q are points on O. The distance from O to PQ is 15 in., and PQ = 16 in. Find the radius of O. . Draw a diagram to represent the situation. The distance from the center of O to PQ is measured along a perpendicular line. . PM = PQ A diameter that is perpendicular to a chord bisects the chord. 1 2 PM = (16) = 8 Substitute. 1 2 OP 2 = PM 2 + OM 2 Use the Pythagorean Theorem. r 2 = 82 + 152 Substitute. r 2 = 289 Simplify. r = 17 Find the square root of each side. Quick Check The radius of O is 17 in. . 12-2

For Exercises 1–5, use the diagram of L. Chords and Arcs Lesson 12-2 Lesson Quiz For Exercises 1–5, use the diagram of L. 1. If YM and ZN are congruent chords, what can you conclude? 2. If YM and ZN are congruent chords, explain why you cannot conclude that LV = LC. 3. Suppose that YM has length 12 in., and its distance from point L is 5 in. Find the radius of L to the nearest tenth. For Exercises 4 and 5, suppose that LV YM, YV = 11 cm, and L has a diameter of 26 cm. 4. Find YM. 5. Find LV to the nearest tenth. . YM ZN; YLM ZLN You do not know whether LV and LC are perpendicular to the chords. 7.8 in. 22 cm 6.9 cm 12-2

hypotenuse divided by 2 : = • = , or 5.5 2 Chords and Arcs Lesson 12-2 Check Skills You’ll Need Solutions 1. The triangle is a 45°-45°-90° triangle, so each leg is the length of the hypotenuse divided by 2 : = • = , or 5.5 2 2. The triangle is a 45°-45°-90° triangle, so each leg is the length of the hypotenuse divided by 2: = 5 3. The triangle is a 30°-60°-90° triangle, so the hypotenuse is twice the length of the side opposite the 30° angle: 2(14) = 28 11 2 11 2 2 11 2 2 5 2 2 12-2