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Geometry - Semester 2 Mrs. Day-Blattner 1/28/2016 Agenda 1/28/2016 1) Turn in Paris trip if finished it at home 2) Race to the Center of the Pool and Circles, Chords, Diameters and their relationships 3) Homework “My trip to Paris” 1. What influenced your choice of location for the hotel? 2. What is the name for the location of the hotel? 3. Why do you think the instructions specified your locations could not be in a line? the point at which the perpendicular bisectors of the sides of a triangle intersect and which is equidistant from the three vertices circumcenter Learning Log (important things to remember) We can only find the circumcenter of 3 points that are not colinear - if we have three points that are co-linear there could not be a location equidistant from all three points of interest. (We could find a point closer to one or 2, but not all 3.) A. B. C. Lesson 4: Central Angles and the chords they subtend. Turn over to… RACE TO THE CENTER OF THE POOL You are about to make some predictions that will reveal how much you already know about the Geometry of circles and chords. The diagram below represents a large circular swimming pool. In the middle of the pool, at a point E, is an elevated lifeguard chair. Chords AB and CD represent catwalks that stretch across the pool. A C E. D B Xena is standing on catwalk AB and Yuliza is on catwalk CD. Xena and Yuliza are identical twins, and they are equally fast swimmers in a race to the lifeguard chair. A C E. D B 5 minutes. You and your partner read questions and answer them as best you can. A C E. B D (pg 19) Lesson 2: Circles, Chords, Diameters, and their Relationships. 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. a point A is said to be equidistant from two different points B and C if AB = AC equidistant On a piece of scratch paper draw 4 circles and then draw a chord of each one. What do you notice? On a piece of scratch paper draw 4 circles and then draw a chord of each one. What do you notice? Now draw the perpendicular bisectors for each chord and extend the line so it goes through the circumference of the circle twice. On a piece of scratch paper draw 4 circles and then draw a chord of each one. What do you notice? Now draw the perpendicular bisectors for each chord and extend the line so it goes through the circumference of the circle twice. The perpendicular bisector of a chord gives 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? 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? It passes through point P, the center of the circle. Draw another chord and label it CD. Construct the perpendicular bisector of CD. What do you notice about the perpendicular bisector of CD? 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? It pass through point P, the center of the circle. Draw another chord and label it CD. Construct the perpendicular bisector of CD. What do you notice about the perpendicular bisector of AB? It passes through point P, the center of the circle. 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. Exercises 1 - 6 (start on page 20) 1. 2. Nick, Ervin, David, Rafi Eleazar, Adrian, Evelyn, Rosa 3. Lorenzo, Sedrik, Nerses, Peter 4. 5. 6. Ani, Leo, Dahsan, Mihail Leslie, Zeke, Harut, Leon Edgar, Georgiy, Melanis, Preny 1.Catrina, Natalie, Aren, Tigran 2. Liam, Michelle, Jake, Kirsten 3. Karlen, Alec, Vahan, Jerry 2 column proof with fully labelled diagram 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. Lesson Summary 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. Homework Problem Set (starting on page 23) 1 - 8. And complete Activity 4. Making and Proving Conjectures about Congruent Chords after clicking on the link below http://projectsharetexas.org/resource/using-logical-reasoning-prove-conjectures-about-circles-ontrack-geometry-module-1lesson-19