Ptolemy's Theorem

Home , Cyclic quadrilateral, Golden ratio, Ptolemy, Right triangle

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M5 GEOMETRY

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.

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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.

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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.

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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.

i. Write a formula for the area of triangle 푩′푨푫 in terms of 풃, 풅, and 풘. Write a formula for the area of triangle 푩′푪푫 in terms of 풂, 풄, and 풘. ퟏ 퐀퐫퐞퐚(푩′푨푫) = 풃풅 ⋅ 퐬퐢퐧(풘) ퟐ and ퟏ 퐀퐫퐞퐚(푩′푪푫) = 풂풄 ⋅ 퐬퐢퐧(풘) ퟐ

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j. Based on the results of part (i), write a formula for the area of cyclic quadrilateral 푨푩푪푫 in terms of 풂, 풃, 풄, 풅, and 풘. ퟏ ퟏ 퐀퐫퐞퐚(푨푩푪푫) = 퐀퐫퐞퐚(푨푩′푪푫) = 풃풅 ⋅ 퐬퐢퐧(풘) + 풂풄 ⋅ 퐬퐢퐧(풘) ퟐ ퟐ

k. Going back to part (a), now establish Ptolemy’s theorem. ퟏ ퟏ ퟏ 푨푪 ⋅ 푩푫 ⋅ 퐬퐢퐧(풘) = 풃풅 ⋅ 퐬퐢퐧(풘) + 풂풄 ⋅ 퐬퐢퐧(풘) The two formulas represent the same area. ퟐ ퟐ ퟐ ퟏ ퟏ ⋅ 퐬퐢퐧(풘) ⋅ (푨푪 ⋅ 푩푫) = ⋅ 퐬퐢퐧(풘) ⋅ (풃풅 + 풂풄) Distributive property ퟐ ퟐ 푨푪 ⋅ 푩푫 = 풃풅 + 풂풄 Multiplicative property of equality

푨푪 ⋅ 푩푫 = (푩푪 ⋅ 푨푫) + (푨푩 ⋅ 푪푫) Substitution

Closing (3 minutes) Gather the class together and ask the following questions: . What was most challenging in your work today?  Answers will vary. Students might say that it was challenging to do the algebra involved or to keep track of congruent angles, for example. . Are you convinced that this theorem holds for all cyclic quadrilaterals?  Answers will vary, but students should say “yes.” . Will Ptolemy’s theorem hold for all quadrilaterals? Explain.  At present, we do not know. The proof seemed very specific to cyclic quadrilaterals, so we might suspect it holds only for these types of quadrilaterals. (If there is time, students can draw an example of a non-cyclic quadrilateral and check that the result does not hold for it.)

Lesson Summary

THEOREM:

PTOLEMY’S THEOREM: For a cyclic quadrilateral 푨푩푪푫, 푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푩푪 ⋅ 푨푫.

Relevant Vocabulary

CYCLIC QUADRILATERAL: A quadrilateral with all vertices lying on a circle is known as a cyclic quadrilateral.

Exit Ticket (5 minutes)

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Name Date

Lesson 21: Ptolemy’s Theorem

Exit Ticket

What is the length of the chord 퐴퐶̅̅̅̅? Explain your answer.

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Exit Ticket Sample Solutions

What is the length of the chord 푨푪̅̅̅̅? Explain your answer.

Chord 푩푫̅̅̅̅̅ is a diameter of the circle because it makes a right angle with chord 푨푪, and 푩푫 = √ퟐퟐ + ퟖퟐ = √ퟔퟖ.

ퟏퟔ√ퟏퟕ 푨푪 ∙ ퟔퟖ = ퟐ ∙ ퟖ + ퟐ ∙ ퟖ 푨푪 = By Ptolemy’s theorem: √ , giving ퟏퟕ .

Problem Set Sample Solutions

1. An equilateral triangle is inscribed in a circle. If 푷 is a point on the circle, what does Ptolemy’s theorem have to say about the distances from this point to the three vertices of the triangle?

It says that the sum of the two shorter distances is equal to the longer distance.

2. Kite 푨푩푪푫 is inscribed in a circle. The kite has an area of ퟏퟎퟖ 퐬퐪. 퐢퐧., and the ratio of the lengths of the non- congruent adjacent sides is ퟑ ∶ ퟏ. What is the perimeter of the kite? ퟏ 푨푪 ⋅ 푩푫 = ퟏퟎퟖ ퟐ 푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푩푪 ⋅ 푨푫 = ퟐퟏퟔ Since 푨푩 = 푩푪 and 푨푫 = 푪푫, then

푨푪 ⋅ 푩푫 = 푨푩 ⋅ 푪푫 + 푨푩 ⋅ 푪푫 ퟐ(푨푩 ⋅ 푪푫) = ퟐퟏퟔ 푨푩 ⋅ 푪푫 = ퟏퟎퟖ.

Let 풙 be the length of 푪푫̅̅̅̅, then

풙 ⋅ ퟑ풙 = ퟏퟎퟖ ퟑ풙ퟐ = ퟏퟎퟖ 풙ퟐ = ퟑퟔ 풙 = ퟔ.

Therefore, 푪푫 and 푨푫 are ퟔ 퐢퐧. and 푨푩 and 푩푪 are ퟏퟖ 퐢퐧., and the perimeter of kite 푨푩푪푫 is ퟒퟖ 퐢퐧.

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3. Draw a right triangle with leg lengths 풂 and 풃 and hypotenuse length 풄. Draw a rotated copy of the triangle such that the figures form a rectangle. What does Ptolemy have to say about this rectangle?

We get 풂ퟐ + 풃ퟐ = 풄ퟐ, the Pythagorean theorem.

4. Draw a regular pentagon of side length ퟏ in a circle. Let 풃 be the length of its diagonals. What does Ptolemy’s theorem say about the quadrilateral formed by four of the vertices of the pentagon?

√ퟓ+ퟏ 풃ퟐ = 풃 + ퟏ 풃 = , so ퟐ . (This is the famous golden ratio!)

5. The area of the inscribed quadrilateral is √ퟑퟎퟎ 퐦퐦ퟐ. Determine the circumference of the circle.

Since 푨푩푪푫 is a rectangle, then 푨푫 ⋅ 푨푩 = √ퟑퟎퟎ, and the diagonals are diameters of the circle.

The length of 푨푩 = ퟐ풓 퐬퐢퐧 ퟔퟎ, and the length of 푨푫 = ퟐ풓 퐬퐢퐧 ퟑퟎ, so 푨푩 = √ퟑ, and 푨푩 = √ퟑ(푨푫). 푨푫 푨푫 ⋅ √ퟑ(푨푫) = √ퟑퟎퟎ √ퟑퟎퟎ 푨푫ퟐ = √ퟑ 푨푫ퟐ = ퟏퟎ 푨푫 = √ퟏퟎ

푨푩 = √ퟑퟎ 푨푩 ⋅ 푫푪 + 푨푫 ⋅ 푩푪 = 푨푪 ⋅ 푩푫 (√ퟑퟎ ⋅ √ퟑퟎ) + (√ퟏퟎ ⋅ √ퟏퟎ) = 푨푪 ⋅ 푨푪 ퟑퟎ + ퟏퟎ = 푨푪ퟐ ퟒퟎ = 푨푪ퟐ ퟐ√ퟏퟎ = 푨푪

Since 푨푪 is ퟐ√ퟏퟎ 퐦퐦, the radius of the circle is √ퟏퟎ 퐦퐦, and the circumference of the circle is ퟐ√ퟏퟎ 흅 퐦퐦.

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6. Extension: Suppose 풙 and 풚 are two acute angles, and the circle has a diameter of ퟏ unit. Find 풂, 풃, 풄, and 풅 in terms of 풙 and 풚. Apply Ptolemy’s theorem, and determine the exact value of 퐬퐢퐧(ퟕퟓ). Use scaffolded questions below as needed.

풂 a. Explain why equals the diameter of the circle. 퐬퐢퐧(풙) If the diameter is ퟏ, this is a right triangle because it is inscribed 풂 풂 in a semicircle, so 퐬퐢퐧(풙) = , or ퟏ = . Since the diameter ퟏ 퐬퐢퐧(풙) 풂 is ퟏ, the diameter is equal to . 퐬퐢퐧(풙)

b. If the circle has a diameter of ퟏ, what is 풂?

풂 = 퐬퐢퐧(풙)

c. Use Thales’ theorem to write the side lengths in the original diagram in terms of 풙 and 풚.

Since both are right triangles, the side lengths are 풂 = 퐬퐢퐧(풙), 풃 = 퐜퐨퐬(풙), 풄 = 퐜퐨퐬(풚), and 풅 = 퐬퐢퐧(풚).

d. If one diagonal of the cyclic quadrilateral is ퟏ, what is the other?

퐬퐢퐧(풙 + 풚)

e. What does Ptolemy’s theorem give?

ퟏ ∙ 퐬퐢퐧(풙 + 풚) = 퐬퐢퐧(풙)퐜퐨퐬(풚) + 퐜퐨퐬(풙)퐬퐢퐧(풚)

f. Using the result from part (e), determine the exact value of 퐬퐢퐧(ퟕퟓ).

ퟏ √ퟐ √ퟑ √ퟐ √ퟐ + √ퟔ 퐬퐢퐧(ퟕퟓ) = 퐬퐢퐧(ퟑퟎ + ퟒퟓ) = 퐬퐢퐧(ퟑퟎ)퐜퐨퐬(ퟒퟓ) + 퐜퐨퐬(ퟑퟎ)퐬퐢퐧(ퟒퟓ) = ∙ + ∙ = ퟐ ퟐ ퟐ ퟐ ퟒ

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