All you need to know about Circles! By: Ms. Erwin Circle Book Notes! All you need to know about Circles! By: Ms. Erwin

Day 1

Tangent Chord VOCAB Inscribed Angle Central Angle Secant

Major Arc Minor Arc A Minor Arc = 𝐴𝐵 Major Arc = 𝐴𝐶𝐵 A Minor Arc is named by two endpoints, and it measures between 0°<𝜃<180° A Major Arc is named by Three points on the arc, and it Measures between 𝜃>180° Major Arc B E C Semicircle= 𝐵𝐷𝐶 We name a circle based on its “Center” E is the Center We would call this circle, “Circle E” A circle measures exactly 𝜃=360° A Semicircle is named by Three points on the arc, and it Measures exactly 𝜃=180° D

Inscribed Quadrilateral Inscribed Quadrilateral: F F 𝑬𝑿 𝟏: VOCAB: <𝐶𝐹𝐷 is an INSCRIBED ANGLE <𝑪𝑭𝑫= 𝟏 𝟐 𝒎 𝑩𝑪 Inscribed Quadrilateral A B C D B C E 𝑉𝑂𝐶𝐴𝐵: <𝐶𝐸𝐷 is a CENTRAL ANGLE <𝑪𝑬𝑫=𝒎 𝑫𝑪 Inscribed Quadrilateral: Opposite Angles are SUPPLEMENTARY! Central Angle 𝑬𝑿 𝟐: D

𝐄𝐗 𝟏: 𝟗𝟎°= 𝒎 𝑪𝑫 If <𝑪𝑭𝑫=𝟒𝟓°, 𝒇𝒊𝒏𝒅 𝒎 𝑪𝑫 <𝑪𝑭𝑫= 𝟏 𝟐 𝒎 𝑪𝑫 C D F If <𝑪𝑭𝑫=𝟒𝟓°, 𝒇𝒊𝒏𝒅 𝒎 𝑪𝑫 𝟒𝟓° <𝑪𝑭𝑫= 𝟏 𝟐 𝒎 𝑪𝑫 𝟒𝟓= 𝟏 𝟐 𝒎 𝑪𝑫 𝟗𝟎°= 𝒎 𝑪𝑫

EX 2: If Opposite angles are supplementary in an inscribed quadrilateral, find the following measure. <𝐷=71° Find the measure of <𝐵 A B C D <D+<B=180 71+<B=180 <B=109°

Day 2

Angle Relationships Angles formed by one Angles formed by CHORDS: SECANT and one TANGENT: Angles formed by CHORDS: <𝐴𝐸𝐵 B A C D A B C D E <𝐴𝐸𝐵= 𝐴𝐵 + 𝐷𝐶 2 <𝐴𝐵𝐶= 𝐴𝐷𝐵 2 Angles formed by two SECANTS: Ex 3: Ex 4: Ex 5: Angle Relationships A C B F G <𝐴𝐵𝐶 <𝐴𝐵𝐶= 𝐴𝐶 − 𝐹𝐺 2

Find the 𝑚<𝐷𝐸𝐴 if 𝑚 𝐷𝐴 =104°, 𝑚 𝐷𝐶 =76°, 𝑎𝑛𝑑 𝑚 𝐴𝐵 =65°, Ex 3: Ex 5: B A C D A B C D E 156° 65° 𝑚 𝐵𝐶 =360°− 104°+65°+76° 104° 𝑚 𝐵𝐶 =115° <𝐷𝐸𝐴= 𝐷𝐴 + 𝐵𝐶 2 𝑚 𝐵𝐷𝐴 =360°−156° 76° <𝐷𝐸𝐴= 104°+115° 2 𝑚 𝐵𝐷𝐴 =204° <𝐷𝐸𝐴=109.5° <𝐶𝐵𝐴= 𝐵𝐷𝐴 2 <𝐶𝐵𝐴= 204° 2 Ex 4: Find the 𝑚<𝐴𝐵𝐶 if 𝑚 𝐴𝐹 =70°, 𝑚 𝐺𝐶 =143°, 𝑎𝑛𝑑 𝑚 𝐴𝐶 =98°, 70° A C B F G 𝑚 𝐹𝐺 =360°− 70°+98°+143° <𝐶𝐵𝐴=102° 𝑚 𝐵𝐶 =49° 98° <𝐴𝐵𝐶= 𝐴𝐶 − 𝐹𝐺 2 143° <𝐴𝐵𝐶= 98°−49° 2 <𝐴𝐵𝐶=24.5°

Day 3-4

Segment Relationships One Secant and one Tangent Intersect! Two Chords Intersect! A B C D E B A C D 𝐴𝐸 ∗ 𝐸𝐶 = 𝐷𝐸 ∗ 𝐸𝐵 𝐴𝐷 ∗ 𝐵𝐷 = ( 𝐶𝐷 ) 2 Segment Relationships Two Secants Intersect! EX 6: EX 7: EX 8: A C B F G 𝐴𝐵 ∗ 𝐹𝐵 = 𝐶𝐵 ∗ 𝐺𝐵

EX 6: Find 𝐴𝐸 if 𝐸𝐵 =8, 𝐷𝐸 =6, and 𝐸𝐶 =12 C B F G A B C D E 𝐴𝐸 ∗ 𝐸𝐶 = 𝐷𝐸 ∗ 𝐸𝐵 𝐴𝐸 ∗12=6∗8 𝐴𝐸 ∗12=48 EX 8: Find 𝐴𝐵 if 𝐶𝐵 =10, 𝐺𝐵 =5, and 𝐹𝐵 =4 𝐴𝐸 =4 𝐴𝐵 ∗ 𝐹𝐵 = 𝐶𝐵 ∗ 𝐺𝐵 EX 7: Find 𝐶𝐷 if 𝐴𝐵 =5, 𝐵𝐷 =4 𝐴𝐵 ∗4=10∗5 𝐴𝐷 ∗ 𝐵𝐷 = ( 𝐶𝐷 ) 2 B A C D 𝐴𝐵 ∗4=50 9∗4= ( 𝐶𝐷 ) 2 𝐴𝐵 =12.5 36= ( 𝐶𝐷 ) 2 6= 𝐶𝐷

If two chords are congruent, then the arcs are congruent B C If two chords are congruent, then the arcs are congruent 𝑖𝑓 𝐴𝐵 ≅𝐶𝐵 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐶𝐵 A C B D E If a radius or diameter is perpendicular to a chord, then the radius bisects the chord, and the arc. 𝑖𝑓 𝐴𝐷 𝑖𝑠 𝑝𝑒𝑟𝑝𝑒𝑛𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝐵𝐶 𝑡ℎ𝑒𝑛 𝐵𝐷 ≅ 𝐶𝐷 and 𝐵𝐸 ≅ 𝐸𝐶 A C B D E F G H I Two chords are congruent, if and only if they are equidistant from the center. 𝐺𝐻 ≅ 𝐵𝐶 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐴𝐸 ≅ 𝐴𝐸

A line is tangent to a circle if and only if it is perpendicular to a radius drawn at the point of tangency 𝐶𝐴 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑐𝑖𝑟𝑐𝑙𝑒 𝑇. 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑎𝑛𝑔𝑒𝑛𝑐𝑦 A C T Z X Y If two segments from the same exterior point are tangent to a circle, then they are congruent 𝑌𝑍 𝑎𝑛𝑑 𝑌𝑋 𝑎𝑟𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒. 𝑇ℎ𝑢𝑠. 𝑌𝑍 ≅ 𝑌𝑋

Day 5

Arc Length Sector Area A piece of the circumference of a circle A slice of the circle bounded by 2 radii and an arc 𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ= 𝑥° 360° ∗2𝜋𝑟 𝑆𝑒𝑐𝑡𝑜𝑟 𝐴𝑟𝑒𝑎= 𝑥° 360° ∗𝜋 𝑟 2 Ex 9: Find the length of arc 𝐸𝐹 in circle S if ES=6 Ex 10: Find the area of the blue sector in Cricle M if AM=5 and 𝐴𝑅 =150° E F S A R M

Ex 9: 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑟𝑐 𝐸𝐹 𝑖𝑛 𝑐𝑖𝑟𝑐𝑙𝑒 𝑆 𝑖𝑓 𝐸𝑆=6𝑖𝑛, 𝑎𝑛𝑑 𝑚<𝐸𝑆𝐹=60° 𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ= 𝑥° 360° ∗2𝜋𝑟 E F S 𝐸𝐹 = 60° 360° ∗2𝜋∗6 𝐸𝐹 =6.3 𝑖𝑛 Ex 10: Find the area of the blue sector in Cricle M if AM=5cm and 𝐴𝑅 =150° 𝑆𝑒𝑐𝑡𝑜𝑟 𝐴𝑟𝑒𝑎= 𝑥° 360° ∗𝜋 𝑟 2 A R M 𝑆𝑒𝑐𝑡𝑜𝑟 𝐴𝑟𝑒𝑎= 150° 360° ∗𝜋 5 2 𝑆𝑒𝑐𝑡𝑜𝑟 𝐴𝑟𝑒𝑎=32.7 𝑐𝑚 2

Day 6

Radian Measure We indicate radian measure using the symbol for a central angle 𝜽 𝑻𝒉𝒆 𝒈𝒓𝒆𝒆𝒌 𝒍𝒆𝒕𝒕𝒆𝒓 𝑻𝒉𝒆𝒕𝒂 Radian measure is the ratio of the arc length, l, to the radius of the circle 𝜽= 𝒍 𝒓 𝟑𝟔𝟎°=𝟐𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔

So if 𝟑𝟔𝟎°=𝟐𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔, then what would 𝟏𝟖𝟎°𝒃𝒆 𝒊𝒏 𝒓𝒂𝒅𝒊𝒂𝒏𝒔? 𝟏𝟖𝟎°=𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 Lets see if we can come up with more radian measures 𝝅 𝟐 𝝅 𝟐 90°= ___________𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝝅 𝟐𝝅 𝟑𝝅 𝟐 270°= ___________𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝟑𝝅 𝟐

Ex 1:

You Try!

You Try!

Equation of a Circle “Center-Radius Form” Any circle can be expressed in the standard form (𝑥−ℎ) 2 + (𝑦−𝑘) 2 = 𝒓 2 Where the center of the circle is at the point ℎ,𝑘 And the radius of the circle is 𝒓

EX: 1 Write the equation of the circle with a center at (3, –3) and a radius of 6. (x – h)2 + (y – k)2 = r 2 Equation of circle (x – 3)2 + (y – (–3))2 = 62 Substitution (x – 3)2 + (y + 3)2 = 36 Simplify. (x – 3)2 + (y + 3)2 = 36

You Try! Write the equation of the circle with a center at (2, –4) and a radius of 4. A. (x – 2)2 + (y + 4)2 = 4 B. (x + 2)2 + (y – 4)2 = 4 C. (x – 2)2 + (y + 4)2 = 16 D. (x + 2)2 + (y – 4)2 = 16

You Try Again! Write the equation of the circle graphed to the right. A. x2 + (y + 3)2 = 3 B. x2 + (y – 3)2 = 3 C. x2 + (y + 3)2 = 9 D. x2 + (y – 3)2 = 9

You Try! Which of the following is the graph of x2 + y2 –10y = 0? A. B. C. D.