# CIRCLES Module 5- Lesson 1 Thales Theorem Connect CIRCLES Module 5- Lesson 1 Thales Theorem Connect the two intersection points and them Athe and B the circle. Use your index card topoints draw

right with theON vertex ON Mark the intersection Repeat the process so there twoangle rightlabel angles circle Choose another point

on theaare circle B A The hypotenuse of a right triangle whose right angle is ON a circle Will pass through the center of the circle making it a diameter of the circle Module 5 p. S 19 Diagram

Explanation of Diagram Theorem or Relationship If A, B, and C are on the circle and = In geometry, Thales's theorem states that if 3 distinct points lie on a circle where the line that connects two of the points is a diameter, then the angle at the third point is a right angle. Module 5 Lesson 1 p. S5

Diagram Explanation of Diagram If A, B, and C lie on a circle and ABC is a right angle, then is a diameter of the circle and passes through the center. Theorem or Relationship The Converse of Thaless Theorem states that if 3 points lie on a circle and at one of the points is a right angle, then the segment that joins the

other two points is a diameter of the circle and passes through the center Module 5 Lesson 1 p. S5 What do we need to know in order to calculate the area of the circle? Find the area of the circle. 2 () 5 8

6 10 25 I DO 2 Radius of the circle How does Thales Theorem help us find the radius? Because A, B and C are on the circle and because is a right angle, we know Using the Pytagorean triple, 6-8-10,

we know that the diameter of the circle is 10. Therefore, the radius is 5. Circles Module 5- Lesson 1 Thales Theorem We Do M5-L1 p. S4 #1 ( a-c) You Do M5-L1 p. S5 #1 Module 5-Lesson 1 p. S5 -S6 Complete #1 6 COMMON ERRORS-TEXTBOOK BLOOPERS A chord is a

line segment Whose endpoints are two points on a circle Vocabulary M5-L1 p. S5 Module 5 Lesson 1 p. S4 #2 a-c 18 144 128 88 26 Given: Prove:

STATEMENTS REASONS 1.) 1.) Given 2.) DP=DE 2.) DP=DE 3.) Draw radii CD and CE 2.) defn. of bisector 2.) of bisector 3.) defn. between any two points there is

exactly one 3.) between anyline. two points there is exactly one line. 3.) Draw radii CD and CE 4.) CD CE 5.) CPCP 7.) 8.) =180 9.) =180 10.) =90 11.) = 90 12.) 4.) All radii of a circle are 5.) reflexive property 5.) reflexive property 6.) SSS Thm

7) CPCTC 7) CPCTC 8.) Angles that form a linear pair 8.) Angles a linear pair have a that sum form of 180. have a sum of 180. 9.) Substitution 9.) Substitution 10.) Algebraic properties 10.) Algebraic properties 11.) transitive property 11.) transitive property 12.) Definition of perpendicular lines 12.) Definition of perpendicular lines Module 5 Lesson 2 p. S9

#1 Diagram Explanation of Diagram Theorem or Relationship If If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. Given: Prove: STATEMENTS REASONS

1.) 1.) Given 2.)Draw radii 2.) 2.) Through Through any any two two points points is is exactly exactly one line. one line. 3.)

3.) All radii of a circle are 4.) 4.) 4.) Reflexive Reflexive Property Property 5.) 5.) 5.) HL HL Theorem Theorem 6.) DP 6.)

6.) CPCTC CPCTC 7.) 7.) 7.) Definition Definition of of aa bisector bisector Module 5-Lesson 2 p.S9 #2 Diagram Explanation of Diagram If

T Theorem or Relationship If a diameter of a circle is perpendicular to a chord, then it bisects the chord. EQUIDISTANCE from a point to a line(segment) is defined to be the length of the PERPENDICULAR SEGMENT that joins the point and line (segment) Given: Prove: STATEMENTS 1.)

GEREASONS 1.) Given 2.) Draw 2.) Between any two points there is exactly one line. 3.) 4.) 3.) All radii of a circle are equal. 4.) SSS 5.) 5.) Definition of an altitude 6.)

6.) CPCTC C Module 5-Lesson 2 p. S10 #3 Diagram Explanation of Diagram If . Theorem or Relationship In a circle or in congruent circles, if two chords are

congruent, then the center is equidistant from the two chords. Given: Prove: STATEMENTS REASONS 1.) 1.) Given 2.) Draw radii 2.) Between any two points there is exactly one line. 3.)

3.) All radii of a circle are equal. 4.) 4.) HL Theorem 5.) 5.) 5.) CPCTC CPCTC 6.) Defn. 6.) 7.) 7.) Segment Addition Postulate 7.) Segment Addition Postulate

8.) 8.) Substitution 9.) 9.) Transitive Property Module 5-Lesson 2 p. S10 #3 Diagram Explanation of Diagram If then Theorem or Relationship

In a circle, if the center of the circle is equidistance to two chords, then the two chords are congruent. Central Angle An angle whose vertex is at the center of a circle Given: Prove: STATEMENTS REASONS

1.) 1.) Given 2.) 3.) 4.) 2.) All radii of a circle are 3.) SAS Theorem 4.) CPCTC 4.) CPCTC Module 5-Lesson 2 p. S11 #5 Diagram Explanation of Diagram

If , Theorem or Relationship In a circle or in congruent circles, if two chords define congruent central angles, then the chords are congruent Given: Prove: STATEMENTS 1.) 2.) 3.) BEA

4.) BEA REASONS 1.) Given 2.) All radii of a circle are 3.) SSS Theorem 4.) CPCTC 4.) CPCTC Module 5-Lesson 2 p. S11 #6 Diagram Explanation of Diagram If Theorem or Relationship

In a circle or in congruent circles, congruent chords define central angles equal in measure Module 5- Lesson 2 p. S12-14 ,#1-9 Diagram Explanation of Diagram Theorem or Relationship Diagram Explanation of Diagram Theorem or Relationship