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Cyclic Quadrilaterals NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY Lesson 20: Cyclic Quadrilaterals Student Outcomes . Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary. 1 . Students derive and apply the area of cyclic quadrilateral 퐴퐵퐶퐷 as 퐴퐶 ⋅ 퐵퐷 ⋅ sin(푤), where 푤 is the measure 2 of the acute angle formed by diagonals 퐴퐶 and 퐵퐷. 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 Ptolemy’s 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. Quadrilateral 퐴퐵퐶퐷 shown in the Opening Exercise is an example of a cyclic quadrilateral. What do you believe the term cyclic means in this case? The vertices 퐴, 퐵, 퐶, and 퐷 lie on a circle. Discuss the following question with a shoulder partner and then share out: What is the relationship of 푥 and 푦 in the diagram? 푥 and 푦 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 249 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY Opening Exercise (5 minutes) Opening Exercise Given cyclic quadrilateral 푨푩푪푫 shown in the diagram, prove that 풙 + 풚 = ퟏퟖퟎ°. Statements Reasons/Explanations 1. 풎푩푪푫̂ + 풎푫푨푩̂ = ퟑퟔퟎ° 1. 푩푪푫̂ and 푫푨푩̂ are non-overlapping arcs that form a complete circle. ퟏ ퟏ 2. (풎푩푪푫̂ + 풎푫푨푩̂ ) = (ퟑퟔퟎ°) 2. Multiplicative property of equality MP.7 ퟐ ퟐ ퟏ ퟏ 3. (풎푩푪푫̂ ) + (풎푫푨푩̂ ) = ퟏퟖퟎ° 3. Distributive property ퟐ ퟐ ퟏ ퟏ 풙 = (풎푩푪푫̂ ) 풚 = (풎푫푨푩̂ ) 4. Inscribed angle theorem 4. ퟐ and ퟐ 5. 풙 + 풚 = ퟏퟖퟎ° 5. Substitution Example (7 minutes) Pose the question below to students before starting Example 1, and ask them to Scaffolding: hypothesize their answers. Remind students that a . The Opening Exercise shows that if a quadrilateral is cyclic, then its opposite triangle can be inscribed in angles are supplementary. Let’s explore the converse relationship. If a a circle or a circle can be quadrilateral has supplementary opposite angles, is the quadrilateral necessarily circumscribed about a a cyclic quadrilateral? triangle. This allows us to draw a circle on three of Yes the four vertices of the . How can we show that your hypothesis is valid? quadrilateral. It is our job Student answers will vary. to show that the fourth vertex also lies on the Example circle. Given quadrilateral 푨푩푪푫 with 풎∠푨 + 풎∠푪 = ퟏퟖퟎ°, prove that quadrilateral 푨푩푪푫 is cyclic; in . Have students create cyclic other words, prove that points 푨, 푩, 푪, and 푫 lie on the same circle. quadrilaterals and measure angles to see patterns. This supports concrete work. 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 250 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY . First, we are given that angles 퐴 and 퐶 are supplementary. What does this mean about angles 퐵 and 퐷, and why? MP.7 The angle sum of a quadrilateral is 360°, and since it is given that angles 퐴 and 퐶 are supplementary, angles 퐵 and 퐷 must then have a sum of 180°. We, of course, can draw a circle through point 퐴, and we can further draw a circle through points 퐴 and 퐵 (infinitely many circles, actually). Can we draw a circle through points 퐴, 퐵, and 퐶? 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 퐴, 퐵, 퐶, and 퐷? Not all quadrilaterals are cyclic (e.g., a non-rectangular parallelogram), so we cannot assume that a circle can be drawn through vertices 퐴, 퐵, 퐶, and 퐷. Where could point 퐷 lie in relation to the circle? 퐷 could lie on the circle, in the interior of the circle, or on the exterior of the circle. To show that 퐷 lies on the circle with 퐴, 퐵, and 퐶, we need to consider the cases where it is not, and show that those cases are impossible. First, let’s consider the case where 퐷 is outside the circle. On the diagram, use a red pencil to locate and label point 퐷′ such that it is outside the circle; then, draw segments 퐶퐷′ and 퐴퐷′. What do you notice about sides ̅퐴퐷̅̅̅̅′ and 퐶퐷̅̅̅̅̅′ if vertex 퐷′ is outside the circle? The sides intersect the circle and are, therefore, secants. Statements Reasons/Explanations 1. 풎∠푨 + 풎∠푪 = ퟏퟖퟎ° 1. Given 2. Assume point 푫′ lies outside the circle determined by 2. Stated assumption for case 1 points 푨, 푩, and 푪. 3. Segments 푪푫′ and 푨푫′ intersect the circle at distinct 3. If the segments intersect the circle at the same points 푷 and 푸; thus, 풎푷푸̂ > ퟎ°. point, then 푫′ lies on the circle, and the stated assumption (Statement 2) is false. ퟏ 풎∠푫′ = (풎푨푩푪̂ − 풎푷푸̂ ) 4. Secant angle theorem: exterior case 4. ퟐ ퟏ 풎∠푩 = (풎푨푷푪̂) 5. Inscribed angle theorem 5. ퟐ 6. 풎∠푨 + 풎∠푩 + 풎∠푪 + 풎∠푫′ = ퟑퟔퟎ° 6. The angle sum of a quadrilateral is ퟑퟔퟎ°. 7. 풎∠푩 + 풎∠푫′ = ퟏퟖퟎ° 7. Substitution (Statements 1 and 6) ퟏ ퟏ 8. (풎푨푷푪̂ ) + (풎푨푩푪̂ − 풎푷푸̂) = ퟏퟖퟎ° 8. Substitution (Statements 4, 5, and 7) ퟐ ퟐ 9. ퟑퟔퟎ° − 풎푨푷푪̂ = 풎푨푩푪̂ 9. 푨푷푪̂ and 푨푩푪̂ are non-overlapping arcs that form a complete circle with a sum of ퟑퟔퟎ°. ퟏ ퟏ 10. (풎푨푷푪̂ ) + ((ퟑퟔퟎ° − 풎푨푷푪̂ ) − 풎푷푸̂) = ퟏퟖퟎ° 10. Substitution (Statements 8 and 9) ퟐ ퟐ ퟏ ퟏ 11. (풎푨푷푪̂ ) + ퟏퟖퟎ° − (풎푨푷푪̂ ) − 풎푷푸̂ = ퟏퟖퟎ° 11. Distributive property ퟐ ퟐ 12. 풎푷푸̂ = ퟎ° 12. Subtraction property of equality 13. 푫′ cannot lie outside the circle. 13. Statement 12 contradicts our stated assumption that 푷 and 푸 are distinct with 풎푷푸̂ > ퟎ° (Statement 3). Lesson 20: Cyclic Quadrilaterals 251 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY Exercise 1 (5 minutes) Students use a similar strategy to show that vertex 퐷 cannot lie inside the circle. Exercises 1. Assume that vertex 푫′′ lies inside the circle as shown in the diagram. Use a similar argument to the Example to show that vertex 푫′′ cannot lie inside the circle. Statements Reasons/Explanations 1. 풎∠푨 + 풎∠푪 = ퟏퟖퟎ° 1. Given 2. Assume point 푫′′ lies inside the circle determined by 2. Stated assumption for case 2 points 푨, 푩, and 푪. 3. 푪푫⃡ ′′ and 푨푫⃡ ′′ intersect the circle at points 푷 and 푸, 3. If the segments intersect the circle at the same respectively; thus, 풎푷푸̂ > ퟎ˚. point, then 푫′′ lies on the circle, and the stated assumption (Statement 2) is false. ퟏ 풎∠푫′′ = (풎푷푸̂ + 풎푨푩푪̂) 4. Secant angle theorem: interior case 4. ퟐ ퟏ 풎∠푩 = (풎푨푷푪̂) 5. Inscribed angle theorem 5. ퟐ 6. 풎∠푨 + 풎∠푩 + 풎∠푪 + 풎∠푫′′ = ퟑퟔퟎ° 6. The angle sum of a quadrilateral is ퟑퟔퟎ°. 7. 풎∠푩 + 풎∠푫′′ = ퟏퟖퟎ° 7. Substitution (Statements 1 and 6) ퟏ ퟏ 8. Substitution (Statements 4, 5, and 7) 8. (풎푨푷푪̂ ) + (풎푷푸̂ + 풎푨푩푪̂ ) = ퟏퟖퟎ° ퟐ ퟐ 9. 풎푨푩푪̂ = ퟑퟔퟎ° − 풎푨푷푪̂ 9. 푨푷푪̂ and 푨푩푪̂ are non-overlapping arcs that form a complete circle with a sum of ퟑퟔퟎ°. ퟏ ퟏ 10. Substitution (Statements 8 and 9) 10. (풎푨푷푪̂ ) + (풎푷푸̂ + (ퟑퟔퟎ° − 풎푨푷푪̂ )) = ퟏퟖퟎ° ퟐ ퟐ ퟏ ퟏ ퟏ 11. Distributive property 11. (풎푨푷푪̂ ) + (풎푷푸̂) + ퟏퟖퟎ° − (풎푨푷푪̂ ) = ퟏퟖퟎ° ퟐ ퟐ ퟐ 12. 풎푷푸̂ = ퟎ° 12. Subtraction property of equality 13. 푫′′ cannot lie inside the circle. 13. Statement 12 contradicts our stated assumption that 푷 and 푸 are distinct with 풎푷푸̂ > ퟎ° (Statement 3).
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