Cyclic Quadrilaterals

Home , Bicentric quadrilateral, Cyclic quadrilateral, Parallelogram, Quadrilateral, Right triangle, Triangle

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

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 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M5-TE-1.3.0 -10.2015

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

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

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).

. In the Example and Exercise 1, we showed that the fourth vertex 퐷 cannot lie outside the circle or inside the circle. What conclusion does this leave us with?  The fourth vertex must then lie on the circle with points 퐴, 퐵, and 퐶.

Lesson 20: Cyclic Quadrilaterals 252

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

. 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.

. Take a moment to discuss with a shoulder partner what this theorem means and how we can use it.  Answers will vary.

Exercises 2–3 (5 minutes) Students now apply the theorem about cyclic quadrilaterals.

2. Quadrilateral 푷푸푹푺 is a cyclic quadrilateral. Explain why △ 푷푸푻 ~ △ 푺푹푻.

Since 푷푸푹푺 is a cyclic quadrilateral, draw a circle on points 푷, 푸, 푹, and 푺. ∠푷푸푺 and ∠푷푹푺 are angles inscribed in the same 푺푷̂ ; therefore, they are equal in measure. Also, ∠푸푷푹 and ∠푸푺푹 are both inscribed in the same 푸푹̂ ; therefore, they are equal in measure. Therefore, by AA criterion for similar triangles, △ 푷푸푻 ~ △ 푺푹푻.

(Students may also use vertical angles relationship at 푻.)

3. A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of 풂, 풃, 풄, and 풅 as shown? ퟏ 퐀퐫퐞퐚 = ⋅ 퐛퐚퐬퐞 ⋅ 퐡퐞퐢퐠퐡퐭 Using the area formula for a triangle ퟐ , the area of the quadrilateral is the sum of the areas of the four right triangular regions. ퟏ ퟏ ퟏ ퟏ 퐀퐫퐞퐚 = 풂풄 + 풄풃 + 풃풅 + 풅풂 ퟐ ퟐ ퟐ ퟐ

OR ퟏ 퐀퐫퐞퐚 = (풂풄 + 풄풃 + 풃풅 + 풅풂) ퟐ

Lesson 20: Cyclic Quadrilaterals 253

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

Discussion (5 minutes) Redraw the cyclic quadrilateral from Exercise 3 as shown in the diagram to the right. . How does this diagram relate to the area(s) that you found in Exercise 3 in terms of 푎, 푏, 푐, and 푑?  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. . What are the lengths of the sides of the large rectangle?  The lengths of the sides of the large rectangle are 푎 + 푏 and 푐 + 푑. . Using the lengths of the large rectangle, what is its area?  Area = (푎 + 푏)(푐 + 푑) . How is the area of the given cyclic quadrilateral related to the area of the large rectangle? 1  The area of the cyclic quadrilateral is [(푎 + 푏)(푐 + 푑)]. 2 . What does this say about the area of a cyclic quadrilateral with perpendicular diagonals?  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 푤°)? Discuss this question with a shoulder partner before beginning Exercise 6.

Exercises 4–5 (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.

ퟏ 풂풃 퐬퐢퐧(풘) 4. Show that the triangle in the diagram has area ퟐ . Draw an altitude to the side with length 풃 as shown in the diagram to form adjacent right triangles. Using right triangle trigonometry, the sine of the acute angle with degree measure 풘 is 풉 퐬퐢퐧(풘) = 풉 풃 풂 , where is the length of the altitude to side . It then follows that 풉 = 풂.

Using the area formula for a triangle: ퟏ 퐀퐫퐞퐚 = × 퐛퐚퐬퐞 × 퐡퐞퐢퐠퐡퐭 ퟐ ퟏ 퐀퐫퐞퐚 = 풃(풂 퐬퐢퐧(풘)) ퟐ ퟏ 퐀퐫퐞퐚 = 풂풃 퐬퐢퐧(풘) ퟐ

Lesson 20: Cyclic Quadrilaterals 254

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

ퟏ (ퟏퟖퟎ − 풘)° 풂풃 퐬퐢퐧(풘) 5. Show that the triangle with obtuse angle has area ퟐ . Draw an altitude to the line that includes the side of the triangle with length 풃. Using right triangle trigonometry: 풉 퐬퐢퐧(풘) = 풉 풂 , where is the length of the altitude drawn. It follows then that 풉 = 풂 퐬퐢퐧(풘).

Using the area formula for a triangle: ퟏ 퐀퐫퐞퐚 = × 퐛퐚퐬퐞 × 퐡퐞퐢퐠퐡퐭 ퟐ ퟏ 퐀퐫퐞퐚 = 풃(풂 퐬퐢퐧(풘)) ퟐ ퟏ 퐀퐫퐞퐚 = 풂풃 퐬퐢퐧(풘) ퟐ

Exercise 6 (5 minutes) 1 Students in pairs apply the area formula Area = 푎푏 sin(푤), first encountered in Lesson 31 of Module 2, to show that 2 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.

ퟏ 퐀퐫퐞퐚 = (풂 + 풃)(풄 + 풅) 퐬퐢퐧(풘) 6. Show that the area of the cyclic quadrilateral shown in the diagram is ퟐ . Using the area formula for a triangle: ퟏ 퐀퐫퐞퐚 = 풂풃 ⋅ 퐬퐢퐧(풘) ퟐ ퟏ 퐀퐫퐞퐚 = 풂풅 ⋅ 퐬퐢퐧(풘) ퟏ ퟐ ퟏ 퐀퐫퐞퐚 = 풂풄 ⋅ 퐬퐢퐧(풘) ퟐ ퟐ ퟏ 퐀퐫퐞퐚 = 풃풄 ⋅ 퐬퐢퐧(풘) ퟑ ퟐ ퟏ 퐀퐫퐞퐚 = 풃풅 ⋅ 퐬퐢퐧(풘) ퟒ ퟐ

퐀퐫퐞퐚퐭퐨퐭퐚퐥 = 퐀퐫퐞퐚ퟏ + 퐀퐫퐞퐚ퟐ + 퐀퐫퐞퐚ퟑ + 퐀퐫퐞퐚ퟒ

ퟏ ퟏ ퟏ ퟏ 퐀퐫퐞퐚 = 풂풅 ⋅ 퐬퐢퐧(풘) + 풂풄 ⋅ 퐬퐢퐧(풘) + 풃풄 ⋅ 퐬퐢퐧(풘) + 풃풅 ⋅ 퐬퐢퐧(풘) 퐭퐨퐭퐚퐥 ퟐ ퟐ ퟐ ퟐ ퟏ 퐀퐫퐞퐚 = [ 퐬퐢퐧(풘)] (풂풅 + 풂풄 + 풃풄 + 풃풅) 퐭퐨퐭퐚퐥 ퟐ

ퟏ 퐀퐫퐞퐚 = [ 퐬퐢퐧(풘)] [(풂 + 풃)(풄 + 풅)] 퐭퐨퐭퐚퐥 ퟐ

ퟏ 퐀퐫퐞퐚 = (풂 + 풃)(풄 + 풅) 퐬퐢퐧(풘) 퐭퐨퐭퐚퐥 ퟐ

Lesson 20: Cyclic Quadrilaterals 255

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

Closing (3 minutes) Ask students to verbally provide answers to the following closing questions based on the lesson: . What angle relationship exists in any cyclic quadrilateral?  Both pairs of opposite angles are supplementary. . If a quadrilateral has one pair of opposite angles supplementary, does it mean that the quadrilateral is cyclic? Why?  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.  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 풂 and 풃 and acute included angle with degree measure 풘: ퟏ 퐀퐫퐞퐚 = 풂풃 ⋅ 퐬퐢퐧(풘). ퟐ

The area of a cyclic quadrilateral 푨푩푪푫 whose diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅ intersect to form an acute or right angle with degree measure 풘: ퟏ 퐀퐫퐞퐚(푨푩푪푫) = ⋅ 푨푪 ⋅ 푩푫 ⋅ 퐬퐢퐧(풘). ퟐ

Relevant Vocabulary CYCLIC QUADRILATERAL: A quadrilateral inscribed in a circle is called a cyclic quadrilateral.

Exit Ticket (5 minutes)

Lesson 20: Cyclic Quadrilaterals 256

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

Name Date

Lesson 20: Cyclic Quadrilaterals

Exit Ticket

1. What value of 푥 guarantees that the quadrilateral shown in the diagram below is cyclic? Explain.

2. Given quadrilateral 퐺퐾퐻퐽, 푚∠퐾퐺퐽 + 푚∠퐾퐻퐽 = 180°, 푚∠퐻푁퐽 = 60°, 퐾푁 = 4, 푁퐽 = 48, 퐺푁 = 8, and 푁퐻 = 24, find the area of quadrilateral 퐺퐾퐻퐽. Justify your answer.

Lesson 20: Cyclic Quadrilaterals 257

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

Exit Ticket Sample Solutions

1. What value of 풙 guarantees that the quadrilateral shown in the diagram below is cyclic? Explain.

ퟒ풙 + ퟓ풙 − ퟗ = ퟏퟖퟎ ퟗ풙 − ퟗ = ퟏퟖퟎ ퟗ풙 = ퟏퟖퟗ 풙 = ퟐퟏ

If 풙 = ퟐퟏ, then the opposite angles shown are supplementary, and any quadrilateral with supplementary opposite angles is cyclic.

2. Given quadrilateral 푮푲푯푱, 풎∠푲푮푱 + 풎∠푲푯푱 = ퟏퟖퟎ°, 풎∠푯푵푱 = ퟔퟎ°, 푲푵 = ퟒ, 푵푱 = ퟒퟖ, 푮푵 = ퟖ, and 푵푯 = ퟐퟒ, find the area of quadrilateral 푮푲푯푱. Justify your answer.

Opposite angles 푲푮푱 and 푲푯푱 are supplementary, so quadrilateral 푮푲푯푱 is cyclic.

The area of a cyclic quadrilateral: ퟏ 퐀퐫퐞퐚 = (ퟒ + ퟒퟖ)(ퟖ + ퟐퟒ) ⋅ 퐬퐢퐧(ퟔퟎ) ퟐ ퟏ √ퟑ 퐀퐫퐞퐚 = (ퟓퟐ)(ퟑퟐ) ⋅ ퟐ ퟐ √ퟑ 퐀퐫퐞퐚 = ⋅ ퟏퟔퟔퟒ ퟒ 퐀퐫퐞퐚 = ퟒퟏퟔ√ퟑ

The area of quadrilateral 푮푲푯푱 is ퟒퟏퟔ√ퟑ 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 10–12 are the most difficult and may be passed over, especially for struggling students.

1. Quadrilateral 푩푫푪푬 is cyclic, 푶 is the center of the circle, and 풎∠푩푶푪 = ퟏퟑퟎ°. Find 풎∠푩푬푪. ퟏ 풎∠푩푫푪 = 풎∠푩푶푪 풎∠푩푫푪 = ퟔퟓ° By the inscribed angle theorem, ퟐ , so . Opposite angles of cyclic quadrilaterals are supplementary, so

풎∠푩푬푪 + ퟔퟓ° = ퟏퟖퟎ°. Thus, 풎∠푩푬푪 = ퟏퟏퟓ°.

Lesson 20: Cyclic Quadrilaterals 258

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

2. Quadrilateral 푭푨푬푫 is cyclic, 푨푿 = ퟖ, 푭푿 = ퟔ, 푿푫 = ퟑ, and 풎∠푨푿푬 = ퟏퟑퟎ°. Find the area of quadrilateral 푭푨푬푫.

Using the two-chord power rule, (푨푿)(푿푫) = (푭푿)(푿푬).

ퟖ(ퟑ) = ퟔ(푿푬); thus, 푿푬 = ퟒ.

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 ퟓퟎ°. ퟏ 퐀퐫퐞퐚 = (ퟖ + ퟑ)(ퟔ + ퟒ) ⋅ 퐬퐢퐧(ퟓퟎ) ퟐ ퟏ 퐀퐫퐞퐚 = (ퟏퟏ)(ퟏퟎ) ⋅ 퐬퐢퐧(ퟓퟎ) ퟐ 퐀퐫퐞퐚 ≈ ퟒퟐ. ퟏ

The area of quadrilateral 푭푨푬푫 is approximately ퟒퟐ. ퟏ square units.

3. In the diagram below, 푩푬̅̅̅̅ ∥ 푪푫̅̅̅̅ and 풎∠푩푬푫 = ퟕퟐ°. Find the value of 풔 and 풕.

Quadrilateral 푩푪푫푬 is cyclic, so opposite angles are supplementary.

풔 + ퟕퟐ° = ퟏퟖퟎ° 풔 = ퟏퟎퟖ°

Parallel chords 푩푬̅̅̅̅ and 푪푫̅̅̅̅ intercept congruent arcs 푪푩̂ and 푬푫̂ . By angle addition, 풎푪푩푬̂ = 풎푩푬푫̂ , so it follows by the inscribed angle theorem that 풔 = 풕 = ퟏퟎퟖ°.

4. In the diagram below, 푩푪̅̅̅̅ is the diameter, 풎∠푩푪푫 = ퟐퟓ°, and 푪푬̅̅̅̅ ≅ 푫푬̅̅̅̅. Find 풎∠푪푬푫 and 풎∠푬푫푪.

Triangle 푩푪푫 is inscribed in a semicircle. By Thales’ theorem, ∠푩푫푪 is a right angle. By the angle sum of a triangle, 풎∠푫푩푪 = ퟔퟓ°. Quadrilateral 푩푪푬푫 is cyclic, so opposite angles are supplementary.

ퟔퟓ° + 풎∠푪푬푫 = ퟏퟖퟎ° 풎∠푪푬푫 = ퟏퟏퟓ°

Triangle 푪푬푫 is isosceles since 푪푬̅̅̅̅ ≅ 푫푬̅̅̅̅, and by base ∠’s, ∠푬푫푪 ≅ ∠푬푪푫.

ퟐ(풎∠푬푫푪) = ퟏퟖퟎ° − ퟏퟏퟓ° by the angle sum of a triangle.

풎∠푬푫푪 = ퟑퟐ. ퟓ°

5. In circle 푨, 풎∠푨푩푫 = ퟏퟓ°. Find 풎∠푩푪푫.

Draw diameter 푩푨푿̅̅̅̅̅̅ such that 푿 is on the circle, and then draw 푫푿̅̅̅̅̅. Triangle 푩푫푿 is inscribed in a semicircle; therefore, angle 푩푫푿 is a right angle. By the angle sum of a triangle, 풎∠푩푿푫 = ퟕퟓ°.

Quadrilateral 푩푪푫푿 is a cyclic quadrilateral, so its opposite angles are supplementary.

풎∠푩푪푫 + 풎∠푩푿푫 = ퟏퟖퟎ° 풎∠푩푪푫 + ퟕퟓ° = ퟏퟖퟎ° 풎∠푩푪푫 = ퟏퟎퟓ°

Lesson 20: Cyclic Quadrilaterals 259

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

6. Given the diagram below, 푶 is the center of the circle. If 풎∠푵푶푷 = ퟏퟏퟐ°, find 풎∠푷푸푬.

Draw point 푿 on major arc 푵푷̂ , and draw chords 푿푵̅̅̅̅̅ and 푿푷̅̅̅̅ to form cyclic ퟏ 푸푵푿푷 풎∠푵푿푷 = 풎∠푵푶푷 quadrilateral . By the inscribed angle theorem, ퟐ , so 풎∠푵푿푷 = ퟓퟔ°.

An exterior angle 푷푸푬 of cyclic quadrilateral 푸푵푿푷 is equal in measure to the angle opposite from vertex 푸, which is ∠푵푿푷. Therefore, 풎∠푷푸푬 = ퟓퟔ°.

7. Given the angle measures as indicated in the diagram below, prove that vertices 푪, 푩, 푬, and 푫 lie on a circle.

Using the angle sum of a triangle, 풎∠푭푩푬 = ퟒퟑ°. Angles 푮푩푪 and 푭푩푬 are vertical angles and, therefore, have the same measure. Angles 푮푩푭 and 푬푩푪 are also vertical angles and have the same measure. Angles at a point sum to ퟑퟔퟎ°, so 풎∠푪푩푬 = 풎∠푭푩푮 = ퟏퟑퟕ°. Since ퟏퟑퟕ + ퟒퟑ = ퟏퟖퟎ, angles 푬푩푪 and 푬푫푪 of quadrilateral 푩푪푫푬 are supplementary. If a quadrilateral has opposite angles that are supplementary, then the quadrilateral is cyclic, which means that vertices 푪, 푩, 푬, and 푫 lie on a circle.

8. In the diagram below, quadrilateral 푱푲푳푴 is cyclic. Find the value of 풏.

Angles 푯푲푱 and 푳푲푱 form a linear pair and are supplementary, so angle 푳푲푱 has measure of ퟗퟖ°.

Opposite angles of a cyclic quadrilateral are supplementary; thus, 풎∠푱푴푳 = 풎∠푱푲푯 = ퟖퟐ°.

Angles 푱푴푳 and 푰푴푳 form a linear pair and are supplementary, so angle 푰푴푳 has measure of ퟗퟖ°. Therefore, 풏 = ퟗퟖ.

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 segment’s 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 260

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

10. The circles in the diagram below intersect at points 푨 and 푩. If 풎∠푭푯푮 = ퟏퟎퟎ° and 풎∠푯푮푬 = ퟕퟎ°, find 풎∠푮푬푭 and 풎∠푬푭푯.

Quadrilaterals 푮푯푩푨 and 푬푭푩푨 are both cyclic since their vertices lie on circles. Opposite angles in cyclic quadrilaterals are supplementary.

풎∠푮푨푩 + 풎∠푭푯푮 = ퟏퟖퟎ° 풎∠푮푨푩 + ퟏퟎퟎ° = ퟏퟖퟎ° 풎∠푮푨푩 = ퟖퟎ°

∠푬푨푩 and ∠푮푨푩 are supplementary since they form a linear pair, so 풎∠푬푨푩 = ퟏퟎퟎ°.

∠푬푨푩 and ∠푬푭푩 are supplementary since they are opposite angles in a cyclic quadrilateral, so 풎∠푬푭푩 = ퟖퟎ°.

Using a similar argument:

풎∠푯푮푬 + 풎∠푯푩푨 = 풎∠푯푩푨 + 풎∠푭푩푨 = 풎∠푭푩푨 + 풎∠푮푬푭 = ퟏퟖퟎ° 풎∠푯푮푬 + 풎∠푯푩푨 = 풎∠푯푩푨 + 풎∠푭푩푨 = 풎∠푭푩푨 + 풎∠푮푬푭 = ퟏퟖퟎ° 풎∠푯푮푬 + 풎∠푮푬푭 = ퟏퟖퟎ° ퟕퟎ° + 풎∠푮푬푭 = ퟏퟖퟎ° 풎∠푮푬푭 = ퟏퟏퟎ°

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 풂, 풃, 풄, and 풅.

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:

(풂 + 풃) + (풄 + 풅) = (풅 + 풂) + (풃 + 풄).

Lesson 20: Cyclic Quadrilaterals 261

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M5 GEOMETRY

12. Quadrilateral 푷푺푹푸 is cyclic such that 푷푸̅̅̅̅ is the diameter of the circle. If ∠푸푹푻 ≅ ∠푸푺푹, prove that ∠푷푻푹 is a right angle, and show that 푺, 푿, 푻, and 푷 lie on a circle.

Angles 푹푸푿 and 푺푸푹 are congruent since they are the same angle. Since it was given that ∠푸푹푻 ≅ ∠푸푺푹, it follows that △ 푸푿푹 ~ △ 푸푹푺 by AA similarity criterion. Corresponding angles in similar figures are congruent, so ∠푸푹푺 ≅ ∠푸푿푹, and by vertical ∠’s, ∠푸푿푹 ≅ ∠푻푿푺.

Quadrilateral 푷푺푹푸 is cyclic, so its opposite angles 푸푷푺 and 푸푹푺 are supplementary. Since ∠푸푿푹 ≅ ∠푻푿푺, it follows that angles 푻푿푺 and 푸푷푺 are supplementary. If a quadrilateral has a pair of opposite angles that are supplementary, then the quadrilateral is cyclic. Thus, quadrilateral 푺푿푻푷 is cyclic.

Angle 푷푺푸 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 푷푻푹 must be supplementary to angle 푷푺푸. Therefore, it is a right angle.

Lesson 20: Cyclic Quadrilaterals 262