Warm Up

Properties of Chords  By the end of class today you will: Be able to identify congruent minor arcs in the same or congruent circles using chords Determine if a chord is a diameter Determine if two chords are equal

Use congruent chords to find arc measures  To determine if two minor arcs are congruent, we need to see if the chords that create them are equal.  Theorem 6.5 – In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Using chords to find arc measure  Think about it like cutting a piece of paper… If we cut a paper circle on the red line, we could easily see the arc measure created by the chord.  What if we cut a stack of paper circles just like the one already cut? Would all the arc measures be the same?  What if after we cut them we spun them around. Would their arc measures still be the same? Because each ‘different’ cut is really the same cut, all the arc measures will be the same so long as the chord lengths are the same

Examples

Now You Try!

Using chords to measure  Next we’ll look at a problem that involves using chords as a way to measure distance in a circle.  The Cell Phone Tower Problem

The cell phone tower problem…  You work for a phone company looking to put in a new cell tower in between two towns. It is your job to determine the location. The tower needs to be where each town will get the same reception. Where would you put the tower so that each town was equidistant from the new tower? In the middle of the two towns!

The cell phone tower problem  Since you did such a great job on the last worksite, your boss has asked you to find the best location for another tower by 3 towns. You still need to put the tower in a location equidistant from each of the 3 towns.  How will you do it??? GEOMETRY!!!!!

The cell phone tower problem To solve this, we’ll need to use another theorem: If a chord is a perpendicular bisector of another chord, the first chord is a diameter. We can use this because all diameters of the same circle meet in the middle! So lets make some diameters using perpendicular chords! The points where the bisectors intersect is the center of the circle, so each town is only a ‘radius’ away. 1 st : draw and connect the ‘towns’ 2 nd : draw perpendicular bisectors of those lines 3 rd : those bisectors are diameters of a circle and they meet in the middle!

Properties of perpendicular bisectors of chords  Theorem 6.7: If a diameter is perpendicular to another chord, then the diameter bisects the chord and its arc.

Theorem 6.7  Theorem 6.7 states that if a diameter is a perpendicular bisector of another chord, then it bisects both the chord and that chord’s arc.

Congruent Chords  So far, we have seen how congruent chords can be used to find congruent arc measures but we need to be able to figure out if they are congruent as well.  Theorem 6.8 – Two chords are congruent if and only if they are equidistant from the center.

Congruent Chords

IMPORTANT!!!  With all things geometry, be sure to draw and LABEL everything!  If you know its congruent, label it so!

Homework:  Pg 201; 1-12 all  Pg 203; 1-12 all