sign in sign up
<<
Home , Circumscribed circle, Inscribed angle

10.4 Inscribed and

EEssentialssential QQuestionuestion How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed related to each other?

An inscribed is an angle whose is on a and whose sides R contain chords of the circle. An arc that lies between two lines, rays, or segments is called an intercepted arc. A is an O Q inscribed polygon when all its vertices lie on a circle. intercepted arc P

Inscribed Angles and Central Angles Work with a partner. Use dynamic software. a. Construct an inscribed angle Sample in a circle. Then construct the corresponding central angle. C ATTENDING TO PRECISION b. Measure both angles. How is the inscribed angle related to To be profi cient in math, its intercepted arc? you need to communicate D precisely with others. c. Repeat parts (a) and (b) several A times. Record your results in a table. Write a conjecture about how an inscribed angle is related B to its intercepted arc.

A Quadrilateral with Inscribed Angles Work with a partner. Use dynamic geometry software. a. Construct a quadrilateral with Sample each vertex on a circle.

b. Measure all four angles. What C relationships do you notice? B c. Repeat parts (a) and (b) several times. Record your results in a table. Then write a conjecture A that summarizes the .

E

D

CCommunicateommunicate YourYour AnswerAnswer 3. How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? 4. Quadrilateral EFGH is inscribed in ⊙C, and m∠E = 80°. What is m∠G? Explain.

Section 10.4 Inscribed Angles and Polygons 553

hhs_geo_pe_1004.indds_geo_pe_1004.indd 553553 11/19/15/19/15 2:362:36 PMPM 10.4 Lesson WWhathat YYouou WWillill LLearnearn Use inscribed angles. Core VocabularyVocabulary Use inscribed polygons. inscribed angle, p. 554 Using Inscribed Angles intercepted arc, p. 554 subtend, p. 554 inscribed polygon, p. 556 CCoreore CConceptoncept , p. 556 Inscribed Angle and Intercepted Arc A An inscribed angle is an angle whose inscribed vertex is on a circle and whose sides contain angle B intercepted chords of the circle. An arc that lies between arc two lines, rays, or segments is called an C intercepted arc. If the endpoints of a ២ ∠B intercepts AC . or arc lie on the sides of an inscribed angle, ២ AC subtends ∠B. then the chord or arc is said to subtend AC subtends ∠B. the angle.

TTheoremheorem 10.10 Measure of an Inscribed Angle Theorem A The measure of an inscribed angle is one-half the measure of its intercepted arc. D C B

1 ២ m∠ADB = — m AB Proof Ex. 37, p. 560 2

The proof of the Measure of an Inscribed Angle Theorem involves three cases.

C C C

Case 1 Center C is on a Case 2 Center C is Case 3 Center C is side of the inscribed angle. inside the inscribed angle. outside the inscribed angle.

Using Inscribed Angles

Find the indicated measure. Q ∠ R a. m T 50° ២ P ° b. mQR 48 T S SOLUTION 1 ២ 1 ∠ = — = — ° = ° a. m T 2 mRS 2 (48 ) 24 ២ = ∠ = ° = ° b. mTQ 2២m R 2 ⋅ 50 100២ ២ Because TQR is a semicircle, mQR = 180° − mTQ = 180° − 100° = 80°.

554 Chapter 10

hhs_geo_pe_1004.indds_geo_pe_1004.indd 554554 11/19/15/19/15 2:362:36 PMPM Finding the Measure of an Intercepted Arc ២ Find mRS and m∠STR. What do you notice T about ∠STR and ∠RUS? S

31° R U SOLUTION From២ the Measure of an Inscribed Angle Theorem, you know that mRS = 2m∠RUS = 2(31°) = 62°.

1 ២ 1 ∠ = — = — ° = ° Also, m STR 2 mRS 2 (62 ) 31 .

So, ∠STR ≅∠RUS.

Example 2 suggests the Inscribed Angles of a Circle Theorem.

TTheoremheorem Theorem 10.11 Inscribed Angles of a Circle Theorem If two inscribed angles of a circle intercept A the same arc, then the angles are congruent. D

C B Proof Ex. 38, p. 560 ∠ADB ≅ ∠ACB

Finding the Measure of an Angle

Given m∠E = 75°, fi nd m∠F. G

E 75°

F H SOLUTION ២ Both ∠E and ∠F intercept GH . So, ∠E ≅ ∠F by the Inscribed Angles of a Circle Theorem.

So, m∠F = m∠E = 75°.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Find the measure of the red arc or angle.

1. H 2. TU 3. Y X ° D 38 72° 90° W Z G F V

Section 10.4 Inscribed Angles and Polygons 555

hhs_geo_pe_1004.indds_geo_pe_1004.indd 555555 11/19/15/19/15 2:362:36 PMPM Using Inscribed Polygons CCoreore CConceptoncept Inscribed Polygon circumscribed A polygon is an inscribed polygon when all inscribed circle its vertices lie on a circle. The circle that polygon contains the vertices is a circumscribed circle.

TTheoremsheorems Theorem 10.12 Inscribed Right Theorem A If a is inscribed in a circle, then the is a of the circle. Conversely, if one side of an D inscribed triangle is a diameter of the circle, then the triangle B is a right triangle and the angle opposite the diameter is the C right angle. m∠ABC = 90° if and only if Proof Ex. 39, p. 560 AC is a diameter of the circle.

Theorem 10.13 Inscribed Quadrilateral Theorem F E A quadrilateral can be inscribed in a circle if and only if its C opposite angles are supplementary. G

D Proof Ex. 40, p. 560; D, E, F, and G lie on ⊙C if and only if BigIdeasMath.com m∠D + m∠F = m∠E + m∠G = 180°.

Using Inscribed Polygons

Find the value of each variable. a. B b. D z° E 120° Q y° 80° F G x° A 2 C SOLUTION — a. AB is a diameter. So, ∠C is a right angle, and m∠C = 90° by the Inscribed Right Triangle Theorem. 2x° = 90° x = 45 The value of x is 45. b. DEFG is inscribed in a circle, so opposite angles are supplementary by the Inscribed Quadrilateral Theorem. m∠D + m∠F = 180° m∠E + m∠G = 180° z + 80 = 180 120 + y = 180 z = 100 y = 60 The value of z is 100 and the value of y is 60.

556 Chapter 10 Circles

hhs_geo_pe_1004.indds_geo_pe_1004.indd 556556 11/19/15/19/15 2:362:36 PMPM Constructing a Inscribed in a Circle

Given ⊙C, construct a square inscribed in a circle.

SOLUTION

Step 1 Step 2 Step 3 A A A

EDC E C D C

B B B

Draw a diameter Construct a bisector Form a square Draw any diameter. Label the Construct the perpendicular bisector Connect points A, D, B, and E to endpoints A and B. of the diameter. Label the points form a square. where it intersects ⊙C as points D and E.

Using a Circumscribed Circle

Your camera has a 90° fi eld of vision, and you want to photograph the front of a statue. You stand at a location in which the front of the statue is all that appears in your camera’s fi eld of vision, as shown. You want to change your location. Where else can you stand so that the front of the statue is all that appears in your camera’s fi eld of vision?

SOLUTION From the Inscribed Right Triangle Theorem, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the front of the statue as a diameter.

The statue fi ts perfectly within your camera’s 90° fi eld of vision from any point on the semicircle in front of the statue.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Find the value of each variable.

4. M 5. C 6. T L S ° c° 10x° x° 40 68° (2c − 6)° B x° y° y° ° 8x° U K 82 A D V

7. In Example 5, explain how to fi nd locations where the left side of the statue is all that appears in your camera’s fi eld of vision.

Section 10.4 Inscribed Angles and Polygons 557

hhs_geo_pe_1004.indds_geo_pe_1004.indd 557557 11/19/15/19/15 2:362:36 PMPM 10.4 Exercises Dynamic Solutions available at BigIdeasMath.com

VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. VOCABULARY If a circle is circumscribed about a polygon, then the polygon is an ______.

2. DIFFERENT WORDS, SAME QUESTION Which is different? B Find “both” answers.

∠ ∠ F Find m ABC. Find m AGC. A C 25° 25° G Find m∠AEC. Find m∠ADC. E D

MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics

In Exercises 3–8, fi nd the indicated measure. In Exercises 11 and 12, fi nd the measure of the red arc (See Examples 1 and 2.) or angle. (See Example 3.)

3. m∠A 4. m∠G 11. E 12. P Q F 45° F B 70° A 51° 40° G H S R 84° G C D ° 120 ២ In Exercises 13–16, fi nd the value of each variable. 5. m∠N 6. mRS (See Example 4.)

R EF L 13. R S 14. m° k° M x° y° 2 ° ° Q 67° 60 60 N 95° D G Q 80° ° 160 S T

២ ២ 7. mVU 8. mWX 15. K 16. J X Y Y 3x° U 54° 110° 30° 2y° W M 4b° 34° T a° Z 75° 110° 3 L V X 130°

In Exercises 9 and 10, name two pairs of congruent 17. ERROR ANALYSIS២ Describe and correct the error in angles. fi nding m BC . 9. B 10. W B C ✗ Z X ២ A A 53° mBC = 53° Y D C

558 Chapter 10 Circles

hhs_geo_pe_1004.indds_geo_pe_1004.indd 558558 11/19/15/19/15 2:362:36 PMPM 18. MODELING WITH MATHEMATICS A carpenter’s REASONING In Exercises 25–30, determine whether a square is an L-shaped tool used to draw right angles. quadrilateral of the given type can always be inscribed You need to cut a circular piece of wood into two inside a circle. Explain your reasoning. semicircles. How can you use the carpenter’s square 25. square 26. to draw a diameter on the circular piece of wood? (See Example 5.) 27. 28.

29. 30. isosceles

31. MODELING WITH MATHEMATICS Three moons, A, B, and C, are in the same circular orbit 100,000 kilometers above the surface of a planet. The planet is 20,000 kilometers in diameter and m∠ABC = 90°. Draw a diagram of the situation. How far is moon A from moon C? MATHEMATICAL CONNECTIONS In Exercises 19–21, fi nd the values of x and y. Then fi nd the measures of the 32. MODELING WITH MATHEMATICS At the movie interior angles of the polygon. theater, you want to choose a seat that has the best viewing angle, so that you can be close to the screen 19. B 20. A and still see the whole screen without moving your A x° eyes. You previously decided that seat F7 has the best 3 9y° 26y° D viewing angle, but this time someone else is already 24y° D 21y° sitting there. Where else can you sit so that your seat 14x° has the same viewing angle as seat F7? Explain. 2x° C 4x° B C movie screen

21. B Row A 6y° Row B Row C 6y° Row D A 2x° Row E Row F F7 C Row G 4x° 33. WRITING A right triangle is inscribed in a circle, and 22. MAKING AN ARGUMENT Your friend claims that the of the circle is given. Explain how to fi nd ∠PTQ ≅ ∠PSQ ≅ ∠PRQ. Is your friend correct? the length of the hypotenuse. Explain your reasoning.

T P 34. HOW DO YOU SEE IT? Let point Y represent your location on the soccer fi eld below. What type of ∠ S angle is AYB if you stand anywhere on the circle except at point A or point B?

R Q

23. CONSTRUCTION Construct an A inscribed in a circle. B 24. CONSTRUCTION The side length of an inscribed regular is equal to the radius of the circumscribed circle. Use this fact to construct a regular hexagon inscribed in a circle.

Section 10.4 Inscribed Angles and Polygons 559

hhs_geo_pe_1004.indds_geo_pe_1004.indd 559559 11/19/15/19/15 2:362:36 PMPM 35. WRITING Explain why the diagonals of a rectangle 39. PROVING A THEOREM The Inscribed Right Triangle inscribed in a circle are of the circle. Theorem (Theorem 10.12) is written as a conditional statement and its converse. Write a plan for proof for each statement. 36. THOUGHT PROVOKING The fi gure shows a circle that is circumscribed about △ABC. Is it possible 40. PROVING A THEOREM Copy and complete the to circumscribe a circle about any triangle? Justify paragraph proof for one part of the Inscribed your answer. Quadrilateral Theorem (Theorem 10.13). C F Given ⊙ C with inscribed E quadrilateral DEFG C A Prove m∠D + m∠F = 180°, G B m∠E + m∠G = 180° D

By២ the Arc Addition Postulate២ (Postulate២ 10.1), 37. PROVING A THEOREM If an angle is inscribed in ⊙Q, mEFG + ____ = 360° and mFGD + mDEF = 360°. the center Q can be on a side of the inscribed angle, ២ = ∠ Using២ the ______២ Theorem, mEDG 2m F, inside the inscribed angle, or outside the inscribed mEFG = 2m∠D, mDEF = 2m∠G, and angle. Prove each case of the Measure of an Inscribed ២ mFGD = 2m∠E. By the Substitution Property Angle Theorem (Theorem 10.10). of Equality, 2m∠D + ___ = 360°, so ___. a. Case 1 A Similarly, ___. Given ∠ ABC is inscribed x° ⊙ 41. CRITICAL THINKING In the diagram, ∠C is a right in Q. C B Let m∠B = x°. Q angle. If you —draw the smallest possible circle— through — C to AB , the circle will intersect AC at J and Center Q lies on BC . — — 1 ២ BC at K. Find the exact length of JK . Prove m∠ABC = — mAC 2 C (Hint២: Show that △AQB is isosceles. Then write mAC in terms of x.) 3 4 A

b. Case 2 Use the diagram and A 5 B D B auxiliary line to write Given Q and Prove statements for 42. CRITICAL THINKING You are G Case 2. Then write a proof. C making a circular cutting board. L To begin, you glue eight 1-inch c. Case 3 Use the diagram and A C boards together, as shown. F H auxiliary line to write Given J Then you draw and cut a circle and Prove statements for D B Q with an 8-inch diameter from Case 3. Then write a proof. M the boards. K — 38. PROVING A THEOREM Write a paragraph proof a. FH is a diameter of the circular cutting board. of the Inscribed Angles of a Circle Theorem Write a proportion relating GJ and JH. State a (Theorem 10.11). First, draw a diagram and write theorem to justify your answer. Given and Prove statements. b. Find FJ, JH, and GJ. What —is the length of the cutting board seam labeled GK ?

MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Reviewing what you learned in previous grades and lessons Solve the equation. Check your solution. (Skills Review Handbook) 1 43. = 44. — = 3x 145 2 x 63 1 = = — − 45. 240 2x 46. 75 2 ( x 30)

560 Chapter 10 Circles

hhs_geo_pe_1004.indds_geo_pe_1004.indd 560560 11/19/15/19/15 2:362:36 PMPM