GeometryGeometry 12.3 Arcs and Chords Geometry

Geometry Geometry Objectives/Assignment Use properties of arcs of circles, as applied. Use properties of chords of circles. Reminder Quiz Tomorrow!!!!!!

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a minor arc, so m = m  MRN = 80 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a major arc, so m = 360 ° – 80 ° = 280 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a semicircle, so m = 180 ° 80 °

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 40 ° + 80° = 120° 40 ° 80 ° 110 °

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 120 ° + 110° = 230° 40 ° 80 ° 110 °

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = 360 ° - m = 360 ° - 230° = 130° 40 ° 80 ° 110 °

Geometry Geometry Theorem 12.6 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.  if and only if 

Geometry Geometry Theorem 12.7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. , 

Geometry Geometry Converse of Theorem 12.7 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. is a diameter of the circle.

Geometry Geometry Ex. 4: Using Theorem 12.6 You can use Theorem 10.4 to find m. Because AD  DC, and . So, m = m 2x = x + 40Substitute x = 40 Subtract x from each side. 2x ° (x + 40) °

Geometry Geometry Finding the Center of a Circle Theorem 12.7 can be used to locate a circle’s center as shown in the next few slides. Step 1: Draw any two chords that are not parallel to each other.

Geometry Geometry Finding the Center of a Circle Step 2: Draw the perpendicular bisector of each chord. These are the diameters.

Geometry Geometry Finding the Center of a Circle Step 3: The perpendicular bisectors intersect at the circle’s center.

Geometry Geometry In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB  CD if and only if EF  EG. Theorem 12.9

Geometry Geometry Ex. 7: Using Theorem 12.9 AB = 8; DE = 8, and CD = 5. Find CF.

Geometry Geometry Ex. 7: Using Theorem 12.9 Because AB and DE are congruent chords, they are equidistant from the center. So CF  CG. To find CG, first find DG. CG  DE, so CG bisects DE. Because DE = 8, DG = =4.

Geometry Geometry Ex. 7: Using Theorem 12.9 Then use DG to find CG. DG = 4 and CD = 5, so ∆CGD is a right triangle. So CG = 3. Finally, use CG to find CF. Because CF  CG, CF = CG = 3

Geometry Geometry Reminders: Quiz after 12.3 Last day for seniors is this Friday, make sure you return your books!

Geometry Geometry Homework: Finish the worksheet 12.3 Last day for seniors is this Friday, make sure you return your books!