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Lesson 1: Thales' Theorem NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY Lesson 1: Thales’ Theorem Classwork Opening Exercise a. Mark points and on the sheet of white paper provided by your teacher. b. Take the colored paper provided, and “push” that paper up between points and on the white sheet. 퐴 퐵 c. Mark on the white paper the location of the corner of the colored paper, using a different color than black. 퐴 퐵 Mark that point . See the example below. C 퐶 A B d. Do this again, pushing the corner of the colored paper up between the black points but at a different angle. Again, mark the location of the corner. Mark this point . e. Do this again and then again, multiple times. Continue to label the points. What curve do the colored points 퐷 ( , , …) seem to trace? 퐶 퐷 Exploratory Challenge Choose one of the colored points ( , , ...) that you marked. Draw the right triangle formed by the line segment connecting the original two points and and that colored point. Draw a rotated copy of the triangle underneath it. 퐶 퐷 Label the acute angles in the original퐴 triangle퐵 as and , and label the corresponding angles in the rotated triangle the same. 푥 푦 Todd says ’ is a rectangle. Maryam says ’ is a quadrilateral, but she’s not sure it’s a rectangle. Todd is right but doesn’t know how to explain himself to Maryam. Can you help him out? 퐴퐵퐶퐶 퐴퐵퐶퐶 a. What composite figure is formed by the two triangles? How would you prove it? i. What is the sum of and ? Why? 푥 푦 Lesson 1: Thales’ Theorem Date: 9/5/14 S.1 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY ii. How do we know that the figure whose vertices are the colored points ( , , …) and points and is a rectangle? 퐶 퐷 퐴 퐵 b. Draw the two diagonals of the rectangle. Where is the midpoint of the segment connecting the two original points and ? Why? 퐴 퐵 c. Label the intersection of the diagonals as point . How does the distance from point to a colored point ( , , …) compare to the distance from to points and ? 푃 푃 퐶 퐷 푃 퐴 퐵 d. Choose another colored point, and construct a rectangle using the same process you followed before. Draw the two diagonals of the new rectangle. How do the diagonals of the new and old rectangle compare? How do you know? e. How does your drawing demonstrate that all the colored points you marked do indeed lie on a circle? Lesson 1: Thales’ Theorem Date: 9/5/14 S.2 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY Example 1 In the Exploratory Challenge, you proved the converse of a famous theorem in geometry. Thales’ theorem states: If , , and are three distinct points on a circle and segment is a diameter of the circle, then is right. 퐴Notice퐵 that,퐶 in the proof in the Exploratory Challenge, you started�퐴퐵��� with a right angle (the corner∠ of퐴퐶퐵 the colored paper) and created a circle. With Thales’ theorem, you must start with the circle, and then create a right angle. Prove Thales’ theorem. a. Draw circle with distinct points , , and on the circle and diameter . Prove that is a right angle. 푃 퐴 퐵 퐶 �퐴퐵��� ∠퐴퐶퐵 b. Draw a third radius ( ). What types of triangles are and ? How do you know? �푃퐶��� △ 퐴푃퐶 △ 퐵푃퐶 c. Using the diagram that you just created, develop a strategy to prove Thales’ theorem. d. Label the base angles of as ° and the bases of as °. Express the measure of in terms of ° and °. △ 퐴푃퐶 푏 △ 퐵푃퐶 푎 ∠퐴퐶퐵 푎 푏 e. How can the previous conclusion be used to prove that is a right angle? ∠퐴퐶퐵 Lesson 1: Thales’ Theorem Date: 9/5/14 S.3 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY Exercises 1–2 1. is a diameter of the circle shown. The radius is 12.5 cm, and = 7 cm. a. Find . �퐴퐵��� 퐴퐶 푚∠퐶 b. Find . 퐴퐵 c. Find . 퐵퐶 2. In the circle shown, is a diameter with center . a. Find . 퐵퐶���� 퐴 푚∠퐷퐴퐵 b. Find . 푚∠퐵퐴퐸 c. Find . 푚∠퐷퐴퐸 Lesson 1: Thales’ Theorem Date: 9/5/14 S.4 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY Lesson Summary THEOREMS: • THALES’ THEOREM: If , , and are three different points on a circle with a diameter, then is a right angle. 퐴 퐵 퐶 �퐴퐵��� ∠퐴퐶퐵 • CONVERSE OF THALES’ THEOREM: If is a right triangle with the right angle, then , , and are three distinct points on a circle with a diameter. ∆퐴퐵퐶 ∠퐶 퐴 퐵 퐶 • Therefore, given distinct points , , and on a circle, is a right triangle with the right angle if �퐴퐵��� and only if is a diameter of the circle. 퐴 퐵 퐶 ∆퐴퐵퐶 ∠퐶 • Given two points and , let point be the midpoint between them. If is a point such that is �퐴퐵��� right, then = = . 퐴 퐵 푃 퐶 ∠퐴퐶퐵 퐵푃 퐴푃 퐶푃 Relevant Vocabulary • CIRCLE: Given a point in the plane and a number > 0, the circle with center and radius is the set of all points in the plane that are distance from the point . 퐶 푟 퐶 푟 • RADIUS: May refer either to the line segment joining the center of a circle with any point on that circle (a 푟 퐶 radius) or to the length of this line segment (the radius). • DIAMETER: May refer either to the segment that passes through the center of a circle whose endpoints lie on the circle (a diameter) or to the length of this line segment (the diameter). • CHORD: Given a circle , and let and be points on . The segment is called a chord of . • CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle. 퐶 푃 푄 퐶 �푃푄��� 퐶 Problem Set 1. , , and are three points on a circle, and angle is a right angle. What’s wrong with the picture below? Explain your reasoning. 퐴 퐵 퐶 퐴퐵퐶 Lesson 1: Thales’ Theorem Date: 9/5/14 S.5 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY 2. Show that there is something mathematically wrong with the picture below. 3. In the figure below, is the diameter of a circle of radius 17 miles. If = 30 miles, what is ? �퐴퐵��� 퐵퐶 퐴퐶 4. In the figure below, is the center of the circle, and is a diameter. 푂 �퐴퐷��� a. Find . b. If = 3 4, what is ? 푚∠퐴푂퐵 푚∠퐴푂퐵 ∶ 푚∠퐶푂퐷 ∶ 푚∠퐵푂퐶 Lesson 1: Thales’ Theorem Date: 9/5/14 S.6 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M5 GEOMETRY 5. is a diameter of a circle, and is another point on the circle. The point lies on the line such that = . Show that = . (Hint: Draw a picture to help you explain your thinking!) �푃푄��� 푀 푅 ⃖푀푄�����⃗ 푅푀 푀푄 푚∠푃푅푀 푚∠푃푄푀 6. Inscribe in a circle of diameter 1 such that is a diameter. Explain why: a. sin( ) = . △ 퐴퐵퐶 �퐴퐶��� b. cos( ) = . ∠퐴 퐵퐶 ∠퐴 퐴퐵 Lesson 1: Thales’ Theorem Date: 9/5/14 S.7 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Lesson 2: Circles, Chords, Diameters, and Their Relationships Classwork Opening Exercise Construct the perpendicular bisector of line segment below (as you did in Module 1). �퐴퐵��� Draw another line that bisects but is not perpendicular to it. List one similarity and one difference�퐴퐵��� between the two bisectors. Lesson 2: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.8 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Exercises 1–6 1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. 2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it bisects the chord. Lesson 2: Circle, Chords, Diameters, and Their Relationships Date: 9/5/14 S.9 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY 3. The distance from the center of a circle to a chord is defined as the length of the perpendicular segment from the center to the chord. Note that, since this perpendicular segment may be extended to create a diameter of the circle, therefore, the segment also bisects the chord, as proved in Exercise 2 above.
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