GSE Geometry: Arc, Angles, and Area. Objectives SWBAT understand that circles are similar IOT find circumference and areas of circles SWBAT apply relationships among inscribed angles, radii, chords, tangents and secants IOT solve for unknown quantities in a circle. SWBAT apply the relationship between central, inscribed, and circumscribed angles IOT solve for unknown quantities in a circle. SWBAT define radian measure and derive the formula for area of a sector IOT solve circle problems.

Group Check 30 80 100 180 100 260

Circumference, Arc Length, Area, and Area of Sectors

An Application Problem for Area

A couple of questions concerning Arcs, Angles, and Measurements Arcs and Angles A circle is divided into three arcs of measures 'x', '2x' and '3x.' What is the value of 'x'?

A couple of questions concerning Arcs, Angles, and Measurements Arcs and Lengths

The distance around a circle Circumference The distance around a circle

Circumference or Twinkle, Twinkle Little Star Circumference Equals 2 pi r or

2 Types of Answers Exact Rounded Pi will be in your answer Type the Pi button on your calculator Toggle your answer Do NOT write Pi in your answer Exact Pi will be in your answer

Guided Practice: Find the exact measure. r = 14 feet d = 15 miles

Guided Practice: Find the approximate measure 33 yd 14.3 mm

Arc Length The distance along the curved line making the arc (NOT a degree amount) 

Arc Measure vs Arc Length Measure will be in degrees. Length will be in a “distance” unit of measure EX:

Arc Length

Guided Practice: Find the Arc Length Round to the nearest hundredths 8m  70

Guided Practice: Find the exact Arc Length. 

Guided Practice: What happens to the arc length if the radius were to be doubled? Halved? 

Guided Practice: Find the perimeter of the region.

The amount of space occupied. Area of Circles The amount of space occupied. r A = pr2 

Find the EXACT area. 8. r = 29 feet 9. d = 44 miles

10 and 11 Find the area. Round to the nearest tenths. 53 cm 7.6 yd

Area of a Sector the region bounded by two radii of the circle and their intercepted arc.

Area of a Sector

Find the area of the sector to the nearest hundredths. Guided Practice Find the area of the sector to the nearest hundredths. R 6 cm 60 Q A  18.85 cm2

Find the exact area of the sector. Guided Practice Find the exact area of the sector. 6 cm 120 7 cm Q R

(Area of sector) – (Area of triangle) Guided Practice Area of minor segment = (Area of sector) – (Area of triangle) R Q 12 yd

Inscribed Angles and Inscribed Quadrilaterals

Central Angle Angle = Arc

Inscribed Angle Angle where the vertex is ON the circle

Inscribed Angle

160 The arc is twice as big as the angle!! 80

120  = 120  = 60  Guided Practice: Find the value of x and y. x

Guided Practice 1. If mJK = 80 and <JMK = 2x – 4, find x. x = 22 2. If m<MKS = 56, find m MS. 112  M Q K S J

Guided Practice: Find the measure of <DOG and <DIG 72˚ If two inscribed angles intercept the same arc, then they are congruent. G O I

If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

Quadrilateral inscribed in a circle: opposite angles are SUPPLEMENTARY

If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. What would be the converse of this statement? diameter

x = 3 5x = 2x + 9 3x = + 9 In J, m<3 = 5x and m<4 = 2x + 9. Q D Guided Practice: In J, m<3 = 5x and m<4 = 2x + 9. Find the value of x. Why can we set these angles equal to each other? 3 Q D J T U 4 5x = 2x + 9 3x = + 9 x = 3

x = 26 4x – 14 = 90 4x = 104 Guided Practice: H K N G In K, GH is a diameter and m<GNH = 4x – 14. Find the value of x. What do we automatically know about this triangle? 4x – 14 = 90 H K G N 4x = 104 x = 26

Guided Practice: Solve for x and z. 85 2x +18 + 22x – 6 = 180 24x +12 = 180 22x – 6 24x = 168 x = 7 z + 85 = 180 z = 95

GSE Geometry: Arc, Angles, and Area. Objectives SWBAT understand that circles are similar IOT find circumference and areas of circles SWBAT apply relationships among inscribed angles, radii, chords, tangents and secants IOT solve for unknown quantities in a circle. SWBAT apply the relationship between central, inscribed, and circumscribed angles IOT solve for unknown quantities in a circle. SWBAT define radian measure and derive the formula for area of a sector IOT solve circle problems.

244 x = 105 y = 100 Group Check 1. Solve for arc ABC 2. Solve for x and y. x = 105 y = 100

Secant and Tangent Angles Vertex is INSIDE OR OUTSIDE the circle

Vertex is INSIDE the Circle NOT at the Center

Guided Practice Solve for x 180 – 88 X 88 84 92 x = 100

x = 89 360 – 89 – 93 – 45 133 Guided Practice Solve for x. 93 xº 45 89 133 x = 89

Vertex is OUTside the Circle

Guided Practice Solve for x. 15° x = 25o 65°

Guided Practice Solve for x. 27° x 70° x = 16

Guided Practice Solve for x. 360 – 260 260° 100 x x = 80

Tune: If You’re Happy and You Know It If the vertex is ON the circle half the arc. <clap, clap> If the vertex is IN the circle half the sum. <clap, clap> But if the vertex is OUTside, then you’re in for a ride, cause it’s half of the difference anyway. <clap, clap>

Exit Ticket: Solve for x 1.) 124◦ 2.) 70◦ x 18◦ x 4.) 3.) 260◦ x 110◦ 20◦ x