PH_Geo_10-6_Circles_and_Arcs

Sec. 10 – 6 Circles and Arcs:

Sec. 10 – 6 Circles and Arcs Objectives: 1) To find the measures of central angles and arcs. 2) To find circumferences and arc lengths.

Slide 2:

T C R D Circle – Set of all points equidistant from a given point Center ** Name the circle by its center. Radius – Is a segment that has one endpt @ the center and the other endpt on the circle. Ex. CD Diameter – A segment that contains the center of a circle & has both endpts on the circle. Ex. TR Central Angle – Is an  whose vertex is the center of the circle. Ex. TCD C ** 360 °

Finding measures of Central s:

Finding measures of Central s 25% 8% 27% 40% A B C D E m BAE = = 40% of 360 = (.40) • 360 = 144 mCAD = 8% of 360 (.08)(360) = 28.8 mDAE = 27% of 360 = 97.2

More Circle terms:

More Circle terms P S R T Arc – Part of a Circle. * Measured in degrees ° Minor Arc – Smaller than a semicircle. (< 180 °) * Named by 2 letters * Arc Measure = measure of central  * Ex: RS Major Arc – Greater than a semicircle. (> 180 °) * Name by 3 letters * Order matters * Ex: RTS * Measure = Central  Semicircle – Half of a Circle. * Name by 3 letters * Ex: TRS = 180 

Arcs Continued:

Arcs Continued A B C Adjacent Arcs – Are arcs of the same circle that have exactly one point in common. Ex: AB and BC mBCA = mBC + mCA Arc Addition!!

Ex 1 : Finding the measures of Arcs:

Ex 1 : Finding the measures of Arcs O B C D A 58  32  mBC = mDB = mAD = mAB = m BOC = mBC + mCD mADC – mCD mABC – mBC 32 = 32 + 58 = 90 = 180 – 58 = 122 = 180 – 32 = 148 32 ° 122 ° 148 °

Circumference:

Circumference Circumference – of a circle is the distance around the circle. C = d or C = 2 r Pi = 3.14 Diameter of circle Radius of Cirlce

Ex. 2: Find the circumference of the following circle.:

Ex. 2: Find the circumference of the following circle. 9cm C = 2 r =2(9cm) =18 cm = 56.5cm

Example 2: Circumference :

Example 2: Circumference The diameter of a bicycle wheel is 22in. To the nearest whole number, how many revolutions does the wheel make when the bicycle travels 100ft? Step 1: Convert diameter to feet. 12in in a foot C = d =(1.83ft) = 5.8ft Step 2 finish the prob 100ft/5.8ft = = 17.2 turns 22/12 = 1.83ft

Back to Arcs!!:

Back to Arcs!! The measure of an arc is in degrees. Arc Length – Is a fraction of a circle’s Circumference. It is the piece of string that would form the part of the circle. A B C

Slide 11:

Length of AB = mAB 360 2 r Measure of the arc. It is in Degrees. The Circumference Ex: An arc of 40  represents 40/360 or 1/9 of the circle. * Which means 1/9 of the Circumfernece.

Find the length of ADB in M.:

Find the length of ADB in M. 18cm 150  A B M D mADB = 210 C = 2 r = 2(18cm) = 113cm Length of ADB = (210/360) • (113cm) = 66cm Length of ADB = mADB/360 • 2 r