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Ptolemy's Theorem NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY Lesson 21: Ptolemy’s Theorem Student Outcomes . Students determine the area of a cyclic quadrilateral as a function of its side lengths and the acute angle formed by its diagonals. Students prove Ptolemy’s theorem, which states that for a cyclic quadrilateral 퐴퐵퐶퐷, 퐴퐶 ⋅ 퐵퐷 = 퐴퐵 ⋅ 퐶퐷 + 퐵퐶 ⋅ 퐴퐷. They explore applications of the result. Lesson Notes In this lesson, students work to understand Ptolemy’s theorem, which says that for a cyclic quadrilateral 퐷, 퐴퐶 ⋅ 퐵퐷 = 퐴퐵 ⋅ 퐶퐷 + 퐵퐶 ⋅ 퐴퐷. As such, this lesson focuses on the properties of quadrilaterals inscribed in circles. Ptolemy’s single result and the proof for it codify many geometric facts; for instance, the Pythagorean theorem (G-GPE.A.1, G-GPE.B.4), area formulas, and trigonometry results. Therefore, it serves as a capstone experience to our year-long study of geometry. Students use the area formulas they established in the previous lesson to prove the theorem. A set square, patty paper, compass, and straightedge are needed to complete the Exploratory Challenge. Classwork Opening (2 minutes) The Pythagorean theorem, credited to the Greek mathematician Pythagoras of Samos (ca. 570–ca. 495 BCE), describes a universal relationship among the sides of a right triangle. Every right triangle (in fact every triangle) can be circumscribed by a circle. Six centuries later, Greek mathematician Claudius Ptolemy (ca. 90–ca. 168 CE) discovered a relationship between the side lengths and the diagonals of any quadrilateral inscribed in a circle. As we shall see, Ptolemy’s result can be seen as an extension of the Pythagorean theorem. Opening Exercise (5 minutes) Students are given the statement of Ptolemy’s theorem and are asked to test the theorem by measuring lengths on specific cyclic quadrilaterals they are asked to draw. Students conduct this work in pairs and then gather to discuss their ideas afterwards in class as a whole. Remind students that careful labeling in their diagram is important. Opening Exercise Ptolemy’s theorem says that for a cyclic quadrilateral 푨푩푪푫, 푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푩푪 ⋅ 푨푫. With ruler and a compass, draw an example of a cyclic quadrilateral. Label its vertices 푨, 푩, 푪, and 푫. MP.7 Draw the two diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅. Lesson 21: Ptolemy’s Theorem 263 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 21 M5 GEOMETRY With a ruler, test whether or not the claim that 푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푩푪 ⋅ 푨푫 seems to hold true. Repeat for a second example of a cyclic quadrilateral. Challenge: Draw a cyclic quadrilateral with one side of length zero. What shape is this cyclic quadrilateral? Does MP.7 Ptolemy’s claim hold true for it? Students will see that the relationship 푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푩푪 ⋅ 푨푫 seems to hold, within measuring error. For a quadrilateral with one side of length zero, the figure is a triangle inscribed in a circle. If the length 푨푩 = ퟎ, then the points 푨 and 푩 coincide, and Ptolemy’s theorem states 푨푪 ⋅ 푨푫 = ퟎ ⋅ 푪푫 + 푨푪 ⋅ 푨푫, which is true. Exploratory Challenge (30 minutes): A Journey to Ptolemy’s Theorem This Exploratory Challenge leads students to a proof of Ptolemy’s theorem. Students should work in pairs. The teacher guides as necessary. Exploratory Challenge: A Journey to Ptolemy’s Theorem The diagram shows cyclic quadrilateral 푨푩푪푫 with diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅ intersecting to form an acute angle with degree measure 풘. 푨푩 = 풂, 푩푪 = 풃, 푪푫 = 풄, and 푫푨 = 풅. a. From the last lesson, what is the area of quadrilateral 푨푩푪푫 in terms of the lengths of its diagonals and the angle 풘? Remember this formula for later. ퟏ 퐀퐫퐞퐚(푨푩푪푫) = 푨푪 ⋅ 푩푫 ⋅ 퐬퐢퐧 풘 ퟐ b. Explain why one of the angles, ∠푩푪푫 or ∠푩푨푫, has a measure less than or equal to ퟗퟎ°. Opposite angles of a cyclic quadrilateral are supplementary. These two angles cannot both have measures greater than ퟗퟎ° (both angles could be equal to ퟗퟎ°). c. Let’s assume that ∠푩푪푫 in our diagram is the angle with a measure less than or equal to ퟗퟎ°. Call its measure 풗 degrees. What is the area of triangle 푩푪푫 in terms Scaffolding: of 풃, 풄, and 풗? What is the area of triangle 푩푨푫 in terms of 풂, 풅, and 풗? What is the . For part (c) of the area of quadrilateral 푨푩푪푫 in terms of 풂, 풃, 풄, 풅, and 풗? Exploratory Challenge, If 풗 represents the degree measure of an acute angle, review Exercises 4 and 5 then ퟏퟖퟎ˚ − 풗 would be the degree measure of angle from Lesson 20 with a 푩푨푫 since opposite angles of a cyclic quadrilateral are supplementary. The area of triangle 푩푪푫 could small, targeted group, ퟏ 풃풄 ⋅ 퐬퐢퐧(풗) which show that the area then be calculated using ퟐ , and the area of triangle 푩푨푫 could be calculated by formula for a triangle, ퟏ ퟏ 1 풂풅 ⋅ 퐬퐢퐧(ퟏퟖퟎ˚ − (ퟏퟖퟎ˚ − 풗)), or 풂풅 ⋅ 퐬퐢퐧(풗). So, Area(퐴퐵퐶) = 푎푏 sin(푐), ퟐ ퟐ 2 the area of the quadrilateral 푨푩푪푫 is the sum of the can be used where 푐 areas of triangles 푩푨푫 and 푩푪푫, which provides the represents an acute angle. following: . ퟏ ퟏ Allow advanced learners 퐀퐫퐞퐚(푨푩푪푫) = 풂풅 ⋅ 퐬퐢퐧 풗 + 풃풄 ⋅ 퐬퐢퐧 풗 . ퟐ ퟐ to work through and struggle with the exploration on their own. Lesson 21: Ptolemy’s Theorem 264 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 21 M5 GEOMETRY d. We now have two different expressions representing the area of the same cyclic quadrilateral 푨푩푪푫. Does it seem to you that we are close to a proof of Ptolemy’s claim? Equating the two expressions gives us a relationship that does, admittedly, use the four side lengths of the quadrilateral and the two diagonal lengths, but we also have terms that involve 퐬퐢퐧(풘) and 퐬퐢퐧(풗). These terms are not part of Ptolemy’s equation. In order to reach Ptolemy’s conclusion, in Exploratory Challenge, parts (e)–(j), students use rigid motions to convert the cyclic quadrilateral 퐴퐵퐶퐷 to a new cyclic quadrilateral of the same area with the same side lengths (but in an alternative order) and with its matching angle 푣 congruent to angle 푤 in the original diagram. Equating the areas of these two cyclic quadrilaterals yields the desired result. Again, have students complete this work in homogeneous pairs or small groups. Offer to help students as needed. Scaffolding: e. Trace the circle and points 푨, 푩, 푪, and 푫 onto a sheet of patty paper. Reflect The argument provided in part triangle 푨푩푪 about the perpendicular bisector of diagonal 푨푪̅̅̅̅. Let 푨′, 푩′, and 푪′ be (e), (ii) follows the previous the images of the points 푨, 푩, and 푪, respectively. lesson. An alternative argument is that the perpendicular bisector of a chord of a circle passes through the center of the circle. Reflecting a circle or points on a circle about the perpendicular bisector of the chord is, therefore, a symmetry of the circle; thus, 퐵 must go to a point 퐵′ on the same circle. i. What does the reflection do with points 푨 and 푪? Because the reflection was done about the perpendicular bisector of the diagonal 푨푪̅̅̅̅, the endpoints of the diagonal are images of each other (i.e., 푨 = 푪′ and 푪 = 푨′). ii. Is it correct to draw 푩′ as on the circle? Explain why or why not. Reflections preserve angle measure. Thus, ∠푨푩′푪 ≅ ∠푨푩푪. Also, ∠푨푩푪 is supplementary to ∠푪푫푨; thus, ∠푨푩′푪 is too. This means that 푨푩′푪푫 is a quadrilateral with one pair (and, hence, both pairs) of opposite angles supplementary. Therefore, it is cyclic. This means that 푩′ lies on the circle that passes through 푨, 푪, and 푫. And this is the original circle. iii. Explain why quadrilateral 푨푩′푪푫 has the same area as quadrilateral 푨푩푪푫. Triangle 푨푩′푪 is congruent to triangle 푨푩푪 by a congruence transformation, so these triangles have the same area. It now follows that the two quadrilaterals have the same area. Lesson 21: Ptolemy’s Theorem 265 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 21 M5 GEOMETRY f. The diagram shows angles having degree measures 풖, 풘, 풙, 풚, and 풛. Find and label any other angles having degree measures 풖, 풘, 풙, 풚, or 풛, and justify your answers. See diagram. Justifications include the inscribed angle theorem and vertical angles. g. Explain why 풘 = 풖 + 풛 in your diagram from part (f). The angle with degree measure 풘 is an exterior angle to a triangle with two remote interior angles 풖 and 풛. It follows that 풘 = 풖 + 풛. h. Identify angles of measures 풖, 풙, 풚, 풛, and 풘 in your diagram of the cyclic quadrilateral 푨푩′푪푫 from part (e). See diagram below.
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