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Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Lesson 20: Cyclic Quadrilaterals Student Outcomes ๏ง Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary. ๏ง Students derive and apply the area of cyclic quadrilateral ๐ด๐ต๐ถ๐ท as ๐ด๐ถ โ ๐ต๐ท โ sin(๐ค), where ๐ค is the measure 1 2 of the acute angle formed by diagonals ๐ด๐ถ and ๐ต๐ท. Lesson Notes In Lessons 20 and 21, students experience a culmination of the skills they learned in the previous lessons and modules to reveal and understand powerful relationships that exist among the angles, chord lengths, and areas of cyclic quadrilaterals. Students apply reasoning with angle relationships, similarity, trigonometric ratios and related formulas, and relationships of segments intersecting circles. They begin exploring the nature of cyclic quadrilaterals and use the lengths of the diagonals of cyclic quadrilaterals to determine their area. Next, students construct the circumscribed circle on three vertices of a quadrilateral (a triangle) and use angle relationships to prove that the fourth vertex must also lie on the circle (G-C.A.3). They then use these relationships and their knowledge of similar triangles and trigonometry to prove Ptolemyโs theorem, which states that the product of the lengths of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the lengths of the opposite sides of the cyclic quadrilateral. Classwork Opening (5 minutes) Students first encountered a cyclic quadrilateral in Lesson 5, Exercise 1, part (a), though it was referred to simply as an inscribed polygon. Begin the lesson by discussing the meaning of a cyclic quadrilateral. ๏ง Quadrilateral ๐ด๐ต๐ถ๐ท shown in the Opening Exercise is an example of a cyclic quadrilateral. What do you believe the term cyclic means in this case? ๏บ ๏ง The vertices ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle. Discuss the following question with a shoulder partner and then share out: What is the relationship of ๐ฅ and ๐ฆ in the diagram? ๏บ ๐ฅ and ๐ฆ must be supplementary since they are inscribed in two adjacent arcs that form a complete circle. Make a clear statement to students that a cyclic quadrilateral is a quadrilateral that is inscribed in a circle. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 249 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Opening Exercise (5 minutes) Opening Exercise Given cyclic quadrilateral ๐จ๐ฉ๐ช๐ซ shown in the diagram, prove that ๐ + ๐ = ๐๐๐°. Statements 1. 2. MP.7 3. Reasons/Explanations ฬ + ๐๐ซ๐จ๐ฉ ฬ = ๐๐๐° ๐๐ฉ๐ช๐ซ ๐ ๐ ๐ ๐ ๐ ฬ + ๐๐ซ๐จ๐ฉ (๐๐ฉ๐ช๐ซ ฬ ) = (๐๐๐°) ๐ ฬ )+ (๐๐ฉ๐ช๐ซ ๐ ๐ ฬ ) = ๐๐๐° (๐๐ซ๐จ๐ฉ 1. ฬ and ๐ซ๐จ๐ฉ ฬ are non-overlapping ๐ฉ๐ช๐ซ arcs that form a complete circle. 2. Multiplicative property of equality 3. Distributive property 4. ๐ ฬ ) and ๐ = ๐ (๐๐ซ๐จ๐ฉ ฬ) ๐ = (๐๐ฉ๐ช๐ซ 4. Inscribed angle theorem 5. ๐ + ๐ = ๐๐๐° 5. Substitution ๐ ๐ Example (7 minutes) Pose the question below to students before starting Example 1, and ask them to hypothesize their answers. ๏ง The Opening Exercise shows that if a quadrilateral is cyclic, then its opposite angles are supplementary. Letโs explore the converse relationship. If a quadrilateral has supplementary opposite angles, is the quadrilateral necessarily a cyclic quadrilateral? ๏บ ๏ง Scaffolding: Yes How can we show that your hypothesis is valid? ๏บ Student answers will vary. Example Given quadrilateral ๐จ๐ฉ๐ช๐ซ with ๐โ ๐จ + ๐โ ๐ช = ๐๐๐°, prove that quadrilateral ๐จ๐ฉ๐ช๐ซ is cyclic; in other words, prove that points ๐จ, ๐ฉ, ๐ช, and ๐ซ lie on the same circle. ๏ง Remind students that a triangle can be inscribed in a circle or a circle can be circumscribed about a triangle. This allows us to draw a circle on three of the four vertices of the quadrilateral. It is our job to show that the fourth vertex also lies on the circle. ๏ง Have students create cyclic quadrilaterals and measure angles to see patterns. This supports concrete work. ๏ง Explain proof by contradiction by presenting a simple proof such as 2 points define a line, and have students try to prove this is not true. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 250 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY ๏ง MP.7 First, we are given that angles ๐ด and ๐ถ are supplementary. What does this mean about angles ๐ต and ๐ท, and why? ๏บ ๏ง We, of course, can draw a circle through point ๐ด, and we can further draw a circle through points ๐ด and ๐ต (infinitely many circles, actually). Can we draw a circle through points ๐ด, ๐ต, and ๐ถ? ๏บ ๏ง ๏ง Not all quadrilaterals are cyclic (e.g., a non-rectangular parallelogram), so we cannot assume that a circle can be drawn through vertices ๐ด, ๐ต, ๐ถ, and ๐ท. Where could point ๐ท lie in relation to the circle? ๏บ ๏ง Three non-collinear points can determine a circle, and since the points were given to be vertices of a quadrilateral, the points are non-collinear, so yes. Can we draw a circle through points ๐ด, ๐ต, ๐ถ, and ๐ท? ๏บ ๏ง The angle sum of a quadrilateral is 360°, and since it is given that angles ๐ด and ๐ถ are supplementary, angles ๐ต and ๐ท must then have a sum of 180°. ๐ท could lie on the circle, in the interior of the circle, or on the exterior of the circle. To show that ๐ท lies on the circle with ๐ด, ๐ต, and ๐ถ, we need to consider the cases where it is not, and show that those cases are impossible. First, letโs consider the case where ๐ท is outside the circle. On the diagram, use a red pencil to locate and label point ๐ทโฒ such that it is outside the circle; then, draw segments ๐ถ๐ทโฒ and ๐ด๐ทโฒ. What do you notice about sides ฬ ฬ ฬ ฬ ฬ ๐ด๐ทโฒ and ฬ ฬ ฬ ฬ ฬ ๐ถ๐ทโฒ if vertex ๐ทโฒ is outside the circle? ๏บ The sides intersect the circle and are, therefore, secants. Statements 1. ๐โ ๐จ + ๐โ ๐ช = ๐๐๐° Reasons/Explanations 1. Given 2. Assume point ๐ซโฒ lies outside the circle determined by points ๐จ, ๐ฉ, and ๐ช. 2. Stated assumption for case 1 3. Segments ๐ช๐ซโฒ and ๐จ๐ซโฒ intersect the circle at distinct ฬ > ๐°. points ๐ท and ๐ธ; thus, ๐๐ท๐ธ 3. If the segments intersect the circle at the same point, then ๐ซโฒ lies on the circle, and the stated assumption (Statement 2) is false. 4. ๐ ฬ โ ๐๐ท๐ธ ฬ) ๐โ ๐ซโฒ = (๐๐จ๐ฉ๐ช ๐ 4. Secant angle theorem: exterior case 5. ๐ ฬ) ๐โ ๐ฉ = (๐๐จ๐ท๐ช ๐ 5. Inscribed angle theorem 6. ๐โ ๐จ + ๐โ ๐ฉ + ๐โ ๐ช + ๐โ ๐ซโฒ = ๐๐๐° 6. The angle sum of a quadrilateral is ๐๐๐°. 7. ๐โ ๐ฉ + ๐โ ๐ซโฒ = ๐๐๐° 7. Substitution (Statements 1 and 6) 8. Substitution (Statements 4, 5, and 7) 9. ฬ and ๐จ๐ฉ๐ช ฬ are non-overlapping arcs that ๐จ๐ท๐ช form a complete circle with a sum of ๐๐๐°. 8. 9. 10. 11. ๐ ๐ ๐ ฬ ) + (๐๐จ๐ฉ๐ช ฬ โ ๐๐ท๐ธ ฬ) = ๐๐๐° (๐๐จ๐ท๐ช ๐ ฬ = ๐๐จ๐ฉ๐ช ฬ ๐๐๐° โ ๐๐จ๐ท๐ช ๐ ๐ ๐ ๐ ๐ ฬ ) + ((๐๐๐° โ ๐๐จ๐ท๐ช ฬ ) โ ๐๐ท๐ธ ฬ) = ๐๐๐° (๐๐จ๐ท๐ช ๐ ๐ ฬ ) + ๐๐๐° โ (๐๐จ๐ท๐ช ฬ ) โ ๐๐ท๐ธ ฬ = ๐๐๐° (๐๐จ๐ท๐ช ๐ 10. Substitution (Statements 8 and 9) 11. Distributive property ฬ = ๐° 12. ๐๐ท๐ธ 12. Subtraction property of equality 13. ๐ซโฒ cannot lie outside the circle. 13. Statement 12 contradicts our stated assumption that ๐ท and ๐ธ are distinct with ฬ > ๐° (Statement 3). ๐๐ท๐ธ Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 251 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Exercise 1 (5 minutes) Students use a similar strategy to show that vertex ๐ท cannot lie inside the circle. Exercises Assume that vertex ๐ซโฒโฒ lies inside the circle as shown in the diagram. Use a similar argument to the Example to show that vertex ๐ซโฒโฒ cannot lie inside the circle. 1. Statements 1. ๐โ ๐จ + ๐โ ๐ช = ๐๐๐° 1. Given 2. Assume point ๐ซโฒโฒ lies inside the circle determined by points ๐จ, ๐ฉ, and ๐ช. 2. Stated assumption for case 2 3. โก โก ๐ช๐ซโฒโฒ and ๐จ๐ซโฒโฒ intersect the circle at points ๐ท and ๐ธ, ฬ > ๐ห. respectively; thus, ๐๐ท๐ธ 3. If the segments intersect the circle at the same point, then ๐ซโฒโฒ lies on the circle, and the stated assumption (Statement 2) is false. 4. ๐ ฬ + ๐๐จ๐ฉ๐ช ฬ) ๐โ ๐ซโฒโฒ = (๐๐ท๐ธ ๐ 4. Secant angle theorem: interior case 5. ๐ ฬ) ๐โ ๐ฉ = (๐๐จ๐ท๐ช ๐ 5. Inscribed angle theorem 6. ๐โ ๐จ + ๐โ ๐ฉ + ๐โ ๐ช + ๐โ ๐ซโฒโฒ = ๐๐๐° 6. The angle sum of a quadrilateral is ๐๐๐°. 7. ๐โ ๐ฉ + ๐โ ๐ซโฒโฒ = ๐๐๐° 7. Substitution (Statements 1 and 6) ๐ 8. Substitution (Statements 4, 5, and 7) ฬ = ๐๐๐° โ ๐๐จ๐ท๐ช ฬ ๐๐จ๐ฉ๐ช 9. ฬ and ๐จ๐ฉ๐ช ฬ are non-overlapping arcs that ๐จ๐ท๐ช form a complete circle with a sum of ๐๐๐°. ๐ 10. Substitution (Statements 8 and 9) 8. 9. 10. 11. ๏ง Reasons/Explanations ๐ ๐ ๐ ๐ ๐ ฬ ) + (๐๐ท๐ธ ฬ + ๐๐จ๐ฉ๐ช ฬ ) = ๐๐๐° (๐๐จ๐ท๐ช ๐ ๐ ฬ ) + (๐๐ท๐ธ ฬ + (๐๐๐° โ ๐๐จ๐ท๐ช ฬ )) = ๐๐๐° (๐๐จ๐ท๐ช ๐ ๐ ๐ ๐ ๐ ฬ ) + (๐๐ท๐ธ ฬ) + ๐๐๐° โ (๐๐จ๐ท๐ช ฬ ) = ๐๐๐° (๐๐จ๐ท๐ช 11. Distributive property ฬ = ๐° 12. ๐๐ท๐ธ 12. Subtraction property of equality 13. ๐ซโฒโฒ cannot lie inside the circle. 13. Statement 12 contradicts our stated assumption that ๐ท and ๐ธ are distinct with ฬ > ๐° (Statement 3). ๐๐ท๐ธ In the Example and Exercise 1, we showed that the fourth vertex ๐ท cannot lie outside the circle or inside the circle. What conclusion does this leave us with? ๏บ The fourth vertex must then lie on the circle with points ๐ด, ๐ต, and ๐ถ. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 252 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY ๏ง In the Opening Exercise, you showed that the opposite angles in a cyclic quadrilateral are supplementary. In the Example and Exercise 1, we showed that if a quadrilateral has supplementary opposite angles, then the vertices must lie on a circle. This confirms the following theorem: THEOREM: A quadrilateral is cyclic if and only if its opposite angles are supplementary. ๏ง Take a moment to discuss with a shoulder partner what this theorem means and how we can use it. ๏บ Answers will vary. Exercises 2โ3 (5 minutes) Students now apply the theorem about cyclic quadrilaterals. 2. Quadrilateral ๐ท๐ธ๐น๐บ is a cyclic quadrilateral. Explain why โณ ๐ท๐ธ๐ป ~ โณ ๐บ๐น๐ป. Since ๐ท๐ธ๐น๐บ is a cyclic quadrilateral, draw a circle on points ๐ท, ๐ธ, ๐น, and ๐บ. ฬ ; therefore, they are โ ๐ท๐ธ๐บ and โ ๐ท๐น๐บ are angles inscribed in the same ๐บ๐ท equal in measure. Also, โ ๐ธ๐ท๐น and โ ๐ธ๐บ๐น are both inscribed in the same ฬ ; therefore, they are equal in measure. Therefore, by AA criterion for ๐ธ๐น similar triangles, โณ ๐ท๐ธ๐ป ~ โณ ๐บ๐น๐ป. (Students may also use vertical angles relationship at ๐ป.) 3. A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of ๐, ๐, ๐, and ๐ as shown? Using the area formula for a triangle ๐๐ซ๐๐ = ๐ โ ๐๐๐ฌ๐ โ ๐ก๐๐ข๐ ๐ก๐ญ, the area of the ๐ quadrilateral is the sum of the areas of the four right triangular regions. ๐๐ซ๐๐ = ๐ ๐ ๐ ๐ ๐๐ + ๐๐ + ๐๐ + ๐ ๐ ๐ ๐ ๐ ๐ OR ๐๐ซ๐๐ = ๐ (๐๐ + ๐๐ + ๐๐ + ๐ ๐) ๐ Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 253 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Discussion (5 minutes) Redraw the cyclic quadrilateral from Exercise 3 as shown in the diagram to the right. ๏ง How does this diagram relate to the area(s) that you found in Exercise 3 in terms of ๐, ๐, ๐, and ๐? ๏บ ๏ง What are the lengths of the sides of the large rectangle? ๏บ ๏ง 1 2 The area of the cyclic quadrilateral is [(๐ + ๐)(๐ + ๐)]. What does this say about the area of a cyclic quadrilateral with perpendicular diagonals? ๏บ ๏ง Area = (๐ + ๐)(๐ + ๐) How is the area of the given cyclic quadrilateral related to the area of the large rectangle? ๏บ ๏ง The lengths of the sides of the large rectangle are ๐ + ๐ and ๐ + ๐. Using the lengths of the large rectangle, what is its area? ๏บ ๏ง Each right triangular region in the cyclic quadrilateral is half of a rectangular region. The area of the quadrilateral is the sum of the areas of the triangles, and also half the area of the sum of the four smaller rectangular regions. The area of a cyclic quadrilateral with perpendicular diagonals is equal to one-half the product of the lengths of its diagonals. Can we extend this to other cyclic quadrilaterals (for instance, cyclic quadrilaterals whose diagonals intersect to form an acute angle ๐ค°)? Discuss this question with a shoulder partner before beginning Exercise 6. Exercises 4โ5 (Optional, 5 minutes) These exercises may be necessary for review of the area of a non-right triangle using one acute angle. Assign these as an additional problem set to the previous lesson because the skills have been taught before. If students demonstrate confidence with the content, go directly to Exercise 6. 4. ๐ ๐ Show that the triangle in the diagram has area ๐๐ ๐ฌ๐ข๐ง(๐). Draw an altitude to the side with length ๐ as shown in the diagram to form adjacent right triangles. Using right triangle trigonometry, the sine of the acute angle with degree measure ๐ is ๐ฌ๐ข๐ง(๐) = ๐ , where ๐ is the length of the altitude to side ๐. ๐ It then follows that ๐ = ๐. Using the area formula for a triangle: ๐ × ๐๐๐ฌ๐ × ๐ก๐๐ข๐ ๐ก๐ญ ๐ ๐ ๐๐ซ๐๐ = ๐(๐ ๐ฌ๐ข๐ง(๐)) ๐ ๐ ๐๐ซ๐๐ = ๐๐ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐ = Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 254 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 5. ๐ ๐ Show that the triangle with obtuse angle (๐๐๐ โ ๐)° has area ๐๐ ๐ฌ๐ข๐ง(๐). Draw an altitude to the line that includes the side of the triangle with length ๐. Using right triangle trigonometry: ๐ฌ๐ข๐ง(๐) = ๐ , where ๐ is the length of the altitude drawn. ๐ It follows then that ๐ = ๐ ๐ฌ๐ข๐ง(๐). Using the area formula for a triangle: ๐ × ๐๐๐ฌ๐ × ๐ก๐๐ข๐ ๐ก๐ญ ๐ ๐ ๐๐ซ๐๐ = ๐(๐ ๐ฌ๐ข๐ง(๐)) ๐ ๐ ๐๐ซ๐๐ = ๐๐ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐ = Exercise 6 (5 minutes) 1 2 Students in pairs apply the area formula Area = ๐๐ sin(๐ค), first encountered in Lesson 31 of Module 2, to show that the area of a cyclic quadrilateral is one-half the product of the lengths of its diagonals and the sine of the acute angle formed by their intersection. 6. ๐ ๐ Show that the area of the cyclic quadrilateral shown in the diagram is ๐๐ซ๐๐ = (๐ + ๐)(๐ + ๐ ) ๐ฌ๐ข๐ง(๐). Using the area formula for a triangle: ๐๐ซ๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐๐ซ๐๐๐ญ๐จ๐ญ๐๐ฅ = ๐๐ซ๐๐๐ + ๐๐ซ๐๐๐ + ๐๐ซ๐๐๐ + ๐๐ซ๐๐๐ ๐๐ซ๐๐๐ญ๐จ๐ญ๐๐ฅ = ๐ ๐ ๐ ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐) + ๐๐ โ ๐ฌ๐ข๐ง(๐) + ๐๐ โ ๐ฌ๐ข๐ง(๐) + ๐๐ โ ๐ฌ๐ข๐ง(๐) ๐ ๐ ๐ ๐ ๐ ๐๐ซ๐๐๐ญ๐จ๐ญ๐๐ฅ = [ ๐ฌ๐ข๐ง(๐)] (๐๐ + ๐๐ + ๐๐ + ๐๐ ) ๐ ๐ ๐๐ซ๐๐๐ญ๐จ๐ญ๐๐ฅ = [ ๐ฌ๐ข๐ง(๐)] [(๐ + ๐)(๐ + ๐ )] ๐ ๐๐ซ๐๐๐ญ๐จ๐ญ๐๐ฅ = ๐ (๐ + ๐)(๐ + ๐ ) ๐ฌ๐ข๐ง(๐) ๐ Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 255 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Closing (3 minutes) Ask students to verbally provide answers to the following closing questions based on the lesson: ๏ง What angle relationship exists in any cyclic quadrilateral? ๏บ ๏ง If a quadrilateral has one pair of opposite angles supplementary, does it mean that the quadrilateral is cyclic? Why? ๏บ ๏ง Both pairs of opposite angles are supplementary. Yes. We proved that if the opposite angles of a quadrilateral are supplementary, then the fourth vertex of the quadrilateral must lie on the circle through the other three vertices. Describe how to find the area of a cyclic quadrilateral using its diagonals. ๏บ The area of a cyclic quadrilateral is one-half the product of the lengths of the diagonals and the sine of the acute angle formed at their intersection. Lesson Summary THEOREMS: Given a convex quadrilateral, the quadrilateral is cyclic if and only if one pair of opposite angles is supplementary. The area of a triangle with side lengths ๐ and ๐ and acute included angle with degree measure ๐: ๐๐ซ๐๐ = ๐ ๐๐ โ ๐ฌ๐ข๐ง(๐). ๐ The area of a cyclic quadrilateral ๐จ๐ฉ๐ช๐ซ whose diagonals ฬ ฬ ฬ ฬ ๐จ๐ช and ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ซ intersect to form an acute or right angle with degree measure ๐: ๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) = ๐ โ ๐จ๐ช โ ๐ฉ๐ซ โ ๐ฌ๐ข๐ง(๐). ๐ Relevant Vocabulary CYCLIC QUADRILATERAL: A quadrilateral inscribed in a circle is called a cyclic quadrilateral. Exit Ticket (5 minutes) Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 256 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Name Date Lesson 20: Cyclic Quadrilaterals Exit Ticket 1. What value of ๐ฅ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain. 2. Given quadrilateral ๐บ๐พ๐ป๐ฝ, ๐โ ๐พ๐บ๐ฝ + ๐โ ๐พ๐ป๐ฝ = 180°, ๐โ ๐ป๐๐ฝ = 60°, ๐พ๐ = 4, ๐๐ฝ = 48, ๐บ๐ = 8, and ๐๐ป = 24, find the area of quadrilateral ๐บ๐พ๐ป๐ฝ. Justify your answer. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 257 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Exit Ticket Sample Solutions 1. What value of ๐ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain. ๐๐ + ๐๐ โ ๐ = ๐๐๐ ๐๐ โ ๐ = ๐๐๐ ๐๐ = ๐๐๐ ๐ = ๐๐ If ๐ = ๐๐, then the opposite angles shown are supplementary, and any quadrilateral with supplementary opposite angles is cyclic. 2. Given quadrilateral ๐ฎ๐ฒ๐ฏ๐ฑ, ๐โ ๐ฒ๐ฎ๐ฑ + ๐โ ๐ฒ๐ฏ๐ฑ = ๐๐๐°, ๐โ ๐ฏ๐ต๐ฑ = ๐๐°, ๐ฒ๐ต = ๐, ๐ต๐ฑ = ๐๐, ๐ฎ๐ต = ๐, and ๐ต๐ฏ = ๐๐, find the area of quadrilateral ๐ฎ๐ฒ๐ฏ๐ฑ. Justify your answer. Opposite angles ๐ฒ๐ฎ๐ฑ and ๐ฒ๐ฏ๐ฑ are supplementary, so quadrilateral ๐ฎ๐ฒ๐ฏ๐ฑ is cyclic. The area of a cyclic quadrilateral: ๐ (๐ + ๐๐)(๐ + ๐๐) โ ๐ฌ๐ข๐ง(๐๐) ๐ ๐ โ๐ ๐๐ซ๐๐ = (๐๐)(๐๐) โ ๐ ๐ ๐๐ซ๐๐ = ๐๐ซ๐๐ = โ๐ โ ๐๐๐๐ ๐ ๐๐ซ๐๐ = ๐๐๐โ๐ The area of quadrilateral ๐ฎ๐ฒ๐ฏ๐ฑ is ๐๐๐โ๐ square units. Problem Set Sample Solutions The problems in this Problem Set get progressively more difficult and require use of recent and prior skills. The length of the Problem Set may be too time consuming for students to complete in its entirety. Problems 10โ12 are the most difficult and may be passed over, especially for struggling students. 1. Quadrilateral ๐ฉ๐ซ๐ช๐ฌ is cyclic, ๐ถ is the center of the circle, and ๐โ ๐ฉ๐ถ๐ช = ๐๐๐°. Find ๐โ ๐ฉ๐ฌ๐ช. ๐ ๐ By the inscribed angle theorem, ๐โ ๐ฉ๐ซ๐ช = ๐โ ๐ฉ๐ถ๐ช, so ๐โ ๐ฉ๐ซ๐ช = ๐๐°. Opposite angles of cyclic quadrilaterals are supplementary, so ๐โ ๐ฉ๐ฌ๐ช + ๐๐° = ๐๐๐°. Thus, ๐โ ๐ฉ๐ฌ๐ช = ๐๐๐°. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 258 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY 2. Quadrilateral ๐ญ๐จ๐ฌ๐ซ is cyclic, ๐จ๐ฟ = ๐, ๐ญ๐ฟ = ๐, ๐ฟ๐ซ = ๐, and ๐โ ๐จ๐ฟ๐ฌ = ๐๐๐°. Find the area of quadrilateral ๐ญ๐จ๐ฌ๐ซ. Using the two-chord power rule, (๐จ๐ฟ)(๐ฟ๐ซ) = (๐ญ๐ฟ)(๐ฟ๐ฌ). ๐(๐) = ๐(๐ฟ๐ฌ); thus, ๐ฟ๐ฌ = ๐. The area of a cyclic quadrilateral is equal to the product of the lengths of the diagonals and the sine of the acute angle formed by them. The acute angle formed by the diagonals is ๐๐°. ๐ (๐ + ๐)(๐ + ๐) โ ๐ฌ๐ข๐ง(๐๐) ๐ ๐ ๐๐ซ๐๐ = (๐๐)(๐๐) โ ๐ฌ๐ข๐ง(๐๐) ๐ ๐๐ซ๐๐ โ ๐๐. ๐ ๐๐ซ๐๐ = The area of quadrilateral ๐ญ๐จ๐ฌ๐ซ is approximately ๐๐. ๐ square units. 3. ฬ ฬ ฬ ฬ and ๐โ ๐ฉ๐ฌ๐ซ = ๐๐°. Find the value of ๐ and ๐. ฬ ฬ ฬ ฬ โฅ ๐ช๐ซ In the diagram below, ๐ฉ๐ฌ Quadrilateral ๐ฉ๐ช๐ซ๐ฌ is cyclic, so opposite angles are supplementary. ๐ + ๐๐° = ๐๐๐° ๐ = ๐๐๐° ฬ ฬ ฬ ฬ intercept congruent arcs ๐ช๐ฉ ฬ and ๐ฌ๐ซ ฬ ฬ ฬ ฬ and ๐ช๐ซ ฬ . By angle Parallel chords ๐ฉ๐ฌ ฬ = ๐๐ฉ๐ฌ๐ซ ฬ , so it follows by the inscribed angle theorem addition, ๐๐ช๐ฉ๐ฌ that ๐ = ๐ = ๐๐๐°. 4. ฬ ฬ ฬ ฬ is the diameter, ๐โ ๐ฉ๐ช๐ซ = ๐๐°, and ๐ช๐ฌ ฬ ฬ ฬ ฬ โ ๐ซ๐ฌ ฬ ฬ ฬ ฬ . Find ๐โ ๐ช๐ฌ๐ซ and ๐โ ๐ฌ๐ซ๐ช. In the diagram below, ๐ฉ๐ช Triangle ๐ฉ๐ช๐ซ is inscribed in a semicircle. By Thalesโ theorem, โ ๐ฉ๐ซ๐ช is a right angle. By the angle sum of a triangle, ๐โ ๐ซ๐ฉ๐ช = ๐๐°. Quadrilateral ๐ฉ๐ช๐ฌ๐ซ is cyclic, so opposite angles are supplementary. ๐๐° + ๐โ ๐ช๐ฌ๐ซ = ๐๐๐° ๐โ ๐ช๐ฌ๐ซ = ๐๐๐° ฬ ฬ ฬ ฬ โ ๐ซ๐ฌ ฬ ฬ ฬ ฬ , and by base โ โs, โ ๐ฌ๐ซ๐ช โ โ ๐ฌ๐ช๐ซ. Triangle ๐ช๐ฌ๐ซ is isosceles since ๐ช๐ฌ ๐(๐โ ๐ฌ๐ซ๐ช) = ๐๐๐° โ ๐๐๐° by the angle sum of a triangle. ๐โ ๐ฌ๐ซ๐ช = ๐๐. ๐° 5. In circle ๐จ, ๐โ ๐จ๐ฉ๐ซ = ๐๐°. Find ๐โ ๐ฉ๐ช๐ซ. ฬ ฬ ฬ ฬ ฬ ฬ such that ๐ฟ is on the circle, and then draw ๐ซ๐ฟ ฬ ฬ ฬ ฬ ฬ . Draw diameter ๐ฉ๐จ๐ฟ Triangle ๐ฉ๐ซ๐ฟ is inscribed in a semicircle; therefore, angle ๐ฉ๐ซ๐ฟ is a right angle. By the angle sum of a triangle, ๐โ ๐ฉ๐ฟ๐ซ = ๐๐°. Quadrilateral ๐ฉ๐ช๐ซ๐ฟ is a cyclic quadrilateral, so its opposite angles are supplementary. ๐โ ๐ฉ๐ช๐ซ + ๐โ ๐ฉ๐ฟ๐ซ = ๐๐๐° ๐โ ๐ฉ๐ช๐ซ + ๐๐° = ๐๐๐° ๐โ ๐ฉ๐ช๐ซ = ๐๐๐° Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 259 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 20 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 6. Given the diagram below, ๐ถ is the center of the circle. If ๐โ ๐ต๐ถ๐ท = ๐๐๐°, find ๐โ ๐ท๐ธ๐ฌ. ฬ , and draw chords ฬ ฬ ฬ ฬ ฬ Draw point ๐ฟ on major arc ๐ต๐ท ๐ฟ๐ต and ฬ ฬ ฬ ฬ ๐ฟ๐ท to form cyclic quadrilateral ๐ธ๐ต๐ฟ๐ท. By the inscribed angle theorem, ๐โ ๐ต๐ฟ๐ท = ๐ ๐โ ๐ต๐ถ๐ท, ๐ so ๐โ ๐ต๐ฟ๐ท = ๐๐°. An exterior angle ๐ท๐ธ๐ฌ of cyclic quadrilateral ๐ธ๐ต๐ฟ๐ท is equal in measure to the angle opposite from vertex ๐ธ, which is โ ๐ต๐ฟ๐ท. Therefore, ๐โ ๐ท๐ธ๐ฌ = ๐๐°. 7. Given the angle measures as indicated in the diagram below, prove that vertices ๐ช, ๐ฉ, ๐ฌ, and ๐ซ lie on a circle. Using the angle sum of a triangle, ๐โ ๐ญ๐ฉ๐ฌ = ๐๐°. Angles ๐ฎ๐ฉ๐ช and ๐ญ๐ฉ๐ฌ are vertical angles and, therefore, have the same measure. Angles ๐ฎ๐ฉ๐ญ and ๐ฌ๐ฉ๐ช are also vertical angles and have the same measure. Angles at a point sum to ๐๐๐°, so ๐โ ๐ช๐ฉ๐ฌ = ๐โ ๐ญ๐ฉ๐ฎ = ๐๐๐°. Since ๐๐๐ + ๐๐ = ๐๐๐, angles ๐ฌ๐ฉ๐ช and ๐ฌ๐ซ๐ช of quadrilateral ๐ฉ๐ช๐ซ๐ฌ are supplementary. If a quadrilateral has opposite angles that are supplementary, then the quadrilateral is cyclic, which means that vertices ๐ช, ๐ฉ, ๐ฌ, and ๐ซ lie on a circle. 8. In the diagram below, quadrilateral ๐ฑ๐ฒ๐ณ๐ด is cyclic. Find the value of ๐. Angles ๐ฏ๐ฒ๐ฑ and ๐ณ๐ฒ๐ฑ form a linear pair and are supplementary, so angle ๐ณ๐ฒ๐ฑ has measure of ๐๐°. Opposite angles of a cyclic quadrilateral are supplementary; thus, ๐โ ๐ฑ๐ด๐ณ = ๐โ ๐ฑ๐ฒ๐ฏ = ๐๐°. Angles ๐ฑ๐ด๐ณ and ๐ฐ๐ด๐ณ form a linear pair and are supplementary, so angle ๐ฐ๐ด๐ณ has measure of ๐๐°. Therefore, ๐ = ๐๐. 9. Do all four perpendicular bisectors of the sides of a cyclic quadrilateral pass through a common point? Explain. Yes. A cyclic quadrilateral has vertices that lie on a circle, which means that the vertices are equidistant from the center of the circle. The perpendicular bisector of a segment is the set of points equidistant from the segmentโs endpoints. Since the center of the circle is equidistant from all of the vertices (endpoints of the segments that make up the sides of the cyclic quadrilateral), the center lies on all four perpendicular bisectors of the quadrilateral. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 260 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY 10. The circles in the diagram below intersect at points ๐จ and ๐ฉ. If ๐โ ๐ญ๐ฏ๐ฎ = ๐๐๐° and ๐โ ๐ฏ๐ฎ๐ฌ = ๐๐°, find ๐โ ๐ฎ๐ฌ๐ญ and ๐โ ๐ฌ๐ญ๐ฏ. Quadrilaterals ๐ฎ๐ฏ๐ฉ๐จ and ๐ฌ๐ญ๐ฉ๐จ are both cyclic since their vertices lie on circles. Opposite angles in cyclic quadrilaterals are supplementary. ๐โ ๐ฎ๐จ๐ฉ + ๐โ ๐ญ๐ฏ๐ฎ = ๐๐๐° ๐โ ๐ฎ๐จ๐ฉ + ๐๐๐° = ๐๐๐° ๐โ ๐ฎ๐จ๐ฉ = ๐๐° โ ๐ฌ๐จ๐ฉ and โ ๐ฎ๐จ๐ฉ are supplementary since they form a linear pair, so ๐โ ๐ฌ๐จ๐ฉ = ๐๐๐°. โ ๐ฌ๐จ๐ฉ and โ ๐ฌ๐ญ๐ฉ are supplementary since they are opposite angles in a cyclic quadrilateral, so ๐โ ๐ฌ๐ญ๐ฉ = ๐๐°. Using a similar argument: ๐โ ๐ฏ๐ฎ๐ฌ + ๐โ ๐ฏ๐ฉ๐จ = ๐โ ๐ฏ๐ฉ๐จ + ๐โ ๐ญ๐ฉ๐จ = ๐โ ๐ญ๐ฉ๐จ + ๐โ ๐ฎ๐ฌ๐ญ = ๐๐๐° ๐โ ๐ฏ๐ฎ๐ฌ + ๐โ ๐ฏ๐ฉ๐จ = ๐โ ๐ฏ๐ฉ๐จ + ๐โ ๐ญ๐ฉ๐จ = ๐โ ๐ญ๐ฉ๐จ + ๐โ ๐ฎ๐ฌ๐ญ = ๐๐๐° ๐โ ๐ฏ๐ฎ๐ฌ + ๐โ ๐ฎ๐ฌ๐ญ = ๐๐๐° ๐๐° + ๐โ ๐ฎ๐ฌ๐ญ = ๐๐๐° ๐โ ๐ฎ๐ฌ๐ญ = ๐๐๐° 11. A quadrilateral is called bicentric if it is both cyclic and possesses an inscribed circle. (See diagram to the right.) a. What can be concluded about the opposite angles of a bicentric quadrilateral? Explain. Since a bicentric quadrilateral must be also be cyclic, its opposite angles must be supplementary. b. Each side of the quadrilateral is tangent to the inscribed circle. What does this tell us about the segments contained in the sides of the quadrilateral? Two tangents to a circle from an exterior point form congruent segments. The distances from a vertex of the quadrilateral to the tangent points where it meets the inscribed circle are equal. c. Based on the relationships highlighted in part (b), there are four pairs of congruent segments in the diagram. Label segments of equal length with ๐, ๐, ๐, and ๐ . See diagram on the right. d. What do you notice about the opposite sides of the bicentric quadrilateral? The sum of the lengths of one pair of opposite sides of the bicentric quadrilateral is equal to the sum of the lengths of the other pair of opposite sides: (๐ + ๐) + (๐ + ๐ ) = (๐ + ๐) + (๐ + ๐). Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 261 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY ฬ ฬ ฬ ฬ is the diameter of the circle. If โ ๐ธ๐น๐ป โ โ ๐ธ๐บ๐น, prove that โ ๐ท๐ป๐น is a 12. Quadrilateral ๐ท๐บ๐น๐ธ is cyclic such that ๐ท๐ธ right angle, and show that ๐บ, ๐ฟ, ๐ป, and ๐ท lie on a circle. Angles ๐น๐ธ๐ฟ and ๐บ๐ธ๐น are congruent since they are the same angle. Since it was given that โ ๐ธ๐น๐ป โ โ ๐ธ๐บ๐น, it follows that โณ ๐ธ๐ฟ๐น ~ โณ ๐ธ๐น๐บ by AA similarity criterion. Corresponding angles in similar figures are congruent, so โ ๐ธ๐น๐บ โ โ ๐ธ๐ฟ๐น, and by vertical โ โs, โ ๐ธ๐ฟ๐น โ โ ๐ป๐ฟ๐บ. Quadrilateral ๐ท๐บ๐น๐ธ is cyclic, so its opposite angles ๐ธ๐ท๐บ and ๐ธ๐น๐บ are supplementary. Since โ ๐ธ๐ฟ๐น โ โ ๐ป๐ฟ๐บ, it follows that angles ๐ป๐ฟ๐บ and ๐ธ๐ท๐บ are supplementary. If a quadrilateral has a pair of opposite angles that are supplementary, then the quadrilateral is cyclic. Thus, quadrilateral ๐บ๐ฟ๐ป๐ท is cyclic. Angle ๐ท๐บ๐ธ is a right angle since it is inscribed in a semi-circle. If a quadrilateral is cyclic, then its opposite angles are supplementary; thus, angle ๐ท๐ป๐น must be supplementary to angle ๐ท๐บ๐ธ. Therefore, it is a right angle. Lesson 20: Cyclic Quadrilaterals This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 262 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.