P DIAMETER: Distance across the circle through its center Also known as the longest chord.
P RADIUS: Distance from the center to point on circle
Formula Radius = ½ diameter or Diameter = 2r
D = ? r = ? D = ?
Secant Line: intersects the circle at exactly TWO points
a LINE that intersects the circle exactly ONE time Tangent Line: Forms a 90°angle with one radius Point of Tangency: The point where the tangent intersects the circle
Name the term that best describes the notation. Secant Radius Diameter Chord Tangent
Central Angles An angle whose vertex is at the center of the circle
P E F D Semicircle : An Arc that equals 180° EDF To name: use 3 letters
THINGS TO KNOW AND REMEMBER ALWAYS A circle has 360 degrees A semicircle has 180 degrees Vertical Angles are CONGRUENT Linear Pairs are SUPPLEMENTARY
Formula measure Arc = measure Central Angle
m AB m ACB m AE A B C Q 96 E = = = 96° 264° 84° Find the measures. EB is a diameter.
Tell me the measure of the following arcs. AC is a diameter. 80 100 40 140 A B C D R m DAB = m BCA = 240 260
Using Properties of Tangents HK and HG are tangent to F. Find HG. HK = HG 5a – 32 = 4 + 2a 3a – 32 = 4 2 segments tangent to from same ext. point segments . Substitute 5a – 32 for HK and 4 + 2a for HG. Subtract 2a from both sides. 3a = 36 a = 12 HG = 4 + 2(12) = 28 Add 32 to both sides. Divide both sides by 3. Substitute 12 for a. Simplify.
Applying Congruent Angles, Arcs, and Chords TV WS. Find mWS. 9n – 11 = 7n n = 22 n = 11 = 88° chords have arcs. Def. of arcs Substitute the given measures. Subtract 7n and add 11 to both sides. Divide both sides by 2. Substitute 11 for n. Simplify. mTV = mWS mWS = 7(11) + 11 TV WS
Example 3B: Applying Congruent Angles, Arcs, and Chords C J, and mGCD mNJM. Find NM. GD = NM arcs have chords. GD NM GCD NJM Def. of chords
Find QR to the nearest tenth. Step 2 Use the Pythagorean Theorem. Step 3 Find QR. PQ = 20 Radii of a are . TQ 2 + PT 2 = PQ 2 TQ = 20 2 TQ 2 = 300 TQ 17.3 QR = 2(17.3) = 34.6 Substitute 10 for PT and 20 for PQ. Subtract 10 2 from both sides. Take the square root of both sides. PS QR, so PS bisects QR. Step 1 Draw radius PQ.
The circle graph shows the types of cuisine available in a city. Find mTRQ
Inscribed Angle Inscribed Angle = intercepted Arc/2
160 80 The inscribed angle is half of the intercepted angle
120 x y Find the value of x and y. = 120 = 60
In J, m 3 = 5x and m 4 = 2x + 9. Find the value of x. 3 Q D J T U 4 5x = 2x + 9 x = 3 3x = + 9
4x – 14 = 90 H K G N Example 4 In K, GH is a diameter and m GNH = 4x – 14. Find the value of x. x = 26 4x = 104
z 2x x x – 6 = 180 x = 7 z + 85 = 180 z = 95 Example 5 Solve for x and z. 22x – 6 24x +12 = x = 168
1. Solve for arc ABC 2. Solve for x and y. 244 x = 105 y = 100
Vertex is INSIDE the Circle NOT at the Center
Ex. 1 Solve for x X 88 84 x = 100 180 – 88 92
Ex. 2 Solve for x. 45 93 xºxº 89 x = – 89 – 93 –
Vertex is OUTside the Circle
x Ex. 3 Solve for x. 65° 15° x = 25
x Ex. 4 Solve for x. 27° 70° x = 16
x Ex. 5 Solve for x. 260° x = 80 360 –
Warm up: Solve for x 18 ◦ 1.) x 124 ◦ 70 ◦ x 2.) 3.) x 260 ◦ 20 ◦ 110 ◦ x 4.)
Circumference, Arc Length, Area, and Area of Sectors
Find the EXACT circumference. 1.r = 14 feet 2.d = 15 miles
Ex 3 and 4: Find the circumference. Round to the nearest tenths.
Arc Length The distance along the curved line making the arc (NOT a degree amount)
Arc Length
Ex 5. Find the Arc Length Round to the nearest hundredths 8m 70
Ex 6. Find the exact Arc Length.
Ex 7. What happens to the arc length if the radius were to be doubled? Halved?
Area of Circles The amount of space occupied. r A = r 2
Find the EXACT area. 8. r = 29 feet 9. d = 44 miles
10 and 11 Find the area. Round to the nearest tenths.
Area of a Sector the region bounded by two radii of the circle and their intercepted arc.
Area of a Sector
Example 12 Find the area of the sector to the nearest hundredths. A cm 2 60 6 cm Q R
Example 13 Find the exact area of the sector. 6 cm 120 7 cm Q R
Area of minor segment = (Area of sector) – (Area of triangle) 12 yd R Q Example 14