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# Download Feb 14 Day 3 Chords and Circles

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Geometry - Semester 2 Mrs. Day-Blattner 2/14/2017 Agenda 2/14/2017 1) 2) 3) 4) 5) 6) 7) 8) Bulletin Recap last time’s work Circles, Chords, Diameters and their relationships - intro Homework Hold Up and follow on activity Exercises pages 18-21 Lesson Summary Exit Ticket page 25 Homework the point at which the perpendicular bisectors of the sides of a triangle intersect and which is equidistant from the three vertices circumcenter the point at which the perpendicular bisectors of the sides of a triangle intersect and which is equidistant from the three vertices This circle circumscribes the triangle - the 3 vertices lie on the circumference circumcenter the point at which the perpendicular bisectors of the sides of a triangle intersect and which is equidistant from the three vertices This circle circumscribes the triangle - the 3 vertices lie on the circumference circumcenter A_____B______C if we have three points that are colinear there could not be a location equidistant from all three points . Lesson 2: Circles, Chords, Diameters, and their Relationships (page 17) Classwork Opening Exercise Construct the perpendicular bisector of the line segment AB below. A B Construct the perpendicular bisector of the line segment AB below. Draw another line that bisects segment AB but is not perpendicular to it. Draw another line that bisects segment AB but is not perpendicular to it. List one similarity and one difference between the two bisectors. Draw another line that bisects segment AB but is not perpendicular to it. List one similarity and one difference between the two bisectors. Similarity: Both lines cut the segment into two congruent parts cut it in half. Difference: Draw another line that bisects segment AB but is not perpendicular to it. List one similarity and one difference between the two bisectors. Difference: All points on the perpendicular bisector are equal distance from A and B, but that is not true for points on the other line. a point A is said to be equidistant from two different points B and C if AB = AC equidistant A B B C A C Hold up your homework - 3 different size circles with 2 congruent chords drawn in each one. Now construct perpendicular bisectors for each chord and extend the line so it goes through the circumference of its circle twice. What do we notice? A What do we notice? The perpendicular bisector of a chord can generate a diameter of the circle - it always goes through the center of the circle. Construct a circle of any radius and identify the center as point P. Draw a chord and label it AB. What do you notice about the perpendicular bisector of AB? What can you say about the points on a circle in relation to the center of the circle? What can you say about the points on a circle in relation to the center of the circle? The center of the circle is equidistant from any two points on the circle. Look at the circles, chords, and perpendicular bisectors created by your neighbors. What statement can you make about the perpendicular bisector of any chord of a circle? Why? Look at the circles, chords, and perpendicular bisectors created by your neighbors. What statement can you make about the perpendicular bisector of any chord of a circle? Why? The perpendicular bisector of any chord must always contain the center of the circle. The center of the circle is equidistant from the two end points of the chord because they lie on the circle. Therefore, the center lies on the perpendicular bisector of the chord. That is, the perpendicular bisector contains the center. How does this relate to the definition of the perpendicular bisector of a line segment? The set of all points equidistant from two given points (endpoints of a line segment) is precisely the set of all points on the perpendicular bisector of the line segment. Exercise Page 18 1st one together. 1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. Given: Circle C with diameter DE, chord AB, and AF = BF Prove: DE is perpendicular to AB Proof version 1. Statements 1. 2. 3. 4. 5. 6. 7. AF = BF FC = FC AC = BC triangle AFC is congruent to triangle BFC measure of angle AFC is equal to measure of angle BFC angles AFC and BFC are right angles Line segment DE is perpendicular to line segment AB Reasons. 1. 2. 3. 4. 5. 6. 7. Given Reflexive property radii of same circle are equal in measure Side-side-side congruency postulate corresponding angles of congruent triangles are equal in measure equal angles that form a linear pair each measure 90 degrees Definition of perpendicular lines Proof version 2. Statements 1. 2. 3. 4. 5. 6. 7. AF = BF AC = BC measure of angle FAC is equal to measure of angle FBC triangles AFC and BFC are congruent measure angle AFC = measure of angle BFC angles AFC and BFC are right angles Line segment DE is perpendicular to line segment AB Reasons. 1. 2. 3. 4. 5. 6. 7. Given radii of same circle are equal in measure base angles of an isosceles triangle are equal in measure SAS Corresponding angles of congruent triangles are equal in measure equal angles that form a linear pair each measure 90 degrees Definition of perpendicular lines 2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it must bisect the chord. Given: Circle C with diameter DE, chord AB, and DE is perpendicular to AB Prove: DE bisects AB 2.Proof . Statements 1. 2. 3. 4. 5. 6. 7. 8. 9. Line segment DE is perpendicular to line segment AB angles AFC and BFC are right angles angle AFC is congruent to angle BFC AC = BC measure of angle FAC is equal to measure of angle FBC measure of angle ACF is equal to measure of angle BCF triangles AFC and BFC are congruent AF = BF Line segment DE bisects line segment AB Reasons. 1. 2. 3. 4. 5. 6. 7. 8. 9. Given Definition of perpendicular lines all right angles are congruent radii of the same circle are equal in measure base angles of isosceles triangles are congruent two angles of triangle are equal in measure, so third angles are equal ASA corresponding sides of congruent triangles are equal in length Definition of segment bisector. 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 , the segment also bisects the chord. A Central angle of a circle is an angle whose vertex is the center of a circle. (Created where 2 radii intercept) 3. Prove the theorem: In a circle, if two chords are congruent, then the center is equidistant from the two chords. Group 1 4. Prove the theorem: In a circle, if the center is equidistant from the two chords, then the two chords are congruent. Group 2 5. Group 3 6. Group 4 Lesson Summary - Read aloud and highlight Theorems about chords and diameters in a circle and their converses: ● If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. ● If a diameter of a circle is perpendicular to a chord, then it bisects the chord. Lesson Summary cont. ● If two chords are congruent, then the center is equidistant from the two chords. ● If the center is equidistant from two chords, then the two chords are congruent. Lesson Summary cont. ● Congruent chords define central angles equal in measure. ● If two chords define central angles equal in measure, then they are congruent. Lesson 2. Circles, Chords, Diameters and Their Relationships. (page 25) ● Find the theorem from the lesson summary on page 22 that justifies how you solve 1 and 2a) and write it out in the space for solving that problem. Then solve it! ● For 2b the circle still has a center P and a radius of 10 and AB is still perpendicular to DE, BUT you do not know the length of AB this time, you are going to calculate what it will be with the new data. Homework Problem Set (starting on page 22) Mandatory: 1 - 5. Go ahead and solve the other problems if you have time and interest.