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

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

2 Day 1

3 Tangent Chord VOCAB Inscribed Angle Central Angle Secant

4 Major Arc Minor Arc A Minor Arc = 𝐴𝐡 Major Arc = 𝐴𝐢𝐡
A Minor Arc is named by two endpoints, and it measures between 0Β°<πœƒ<180Β° A Major Arc is named by Three points on the arc, and it Measures between πœƒ>180Β° Major Arc B E C Semicircle= 𝐡𝐷𝐢 We name a circle based on its β€œCenter” E is the Center We would call this circle, β€œCircle E” A circle measures exactly πœƒ=360Β° A Semicircle is named by Three points on the arc, and it Measures exactly πœƒ=180Β° D

5 Inscribed Quadrilateral Inscribed Quadrilateral:
F F 𝑬𝑿 𝟏: VOCAB: <𝐢𝐹𝐷 is an INSCRIBED ANGLE <π‘ͺ𝑭𝑫= 𝟏 𝟐 π’Ž 𝑩π‘ͺ Inscribed Quadrilateral A B C D B C E 𝑉𝑂𝐢𝐴𝐡: <𝐢𝐸𝐷 is a CENTRAL ANGLE <π‘ͺ𝑬𝑫=π’Ž 𝑫π‘ͺ Inscribed Quadrilateral: Opposite Angles are SUPPLEMENTARY! Central Angle 𝑬𝑿 𝟐: D

6 𝐄𝐗 𝟏: πŸ—πŸŽΒ°= π’Ž π‘ͺ𝑫 If <π‘ͺ𝑭𝑫=πŸ’πŸ“Β°, π’‡π’Šπ’π’… π’Ž π‘ͺ𝑫 <π‘ͺ𝑭𝑫= 𝟏 𝟐 π’Ž π‘ͺ𝑫
C D F If <π‘ͺ𝑭𝑫=πŸ’πŸ“Β°, π’‡π’Šπ’π’… π’Ž π‘ͺ𝑫 πŸ’πŸ“Β° <π‘ͺ𝑭𝑫= 𝟏 𝟐 π’Ž π‘ͺ𝑫 πŸ’πŸ“= 𝟏 𝟐 π’Ž π‘ͺ𝑫 πŸ—πŸŽΒ°= π’Ž π‘ͺ𝑫

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

8 Day 2

9 Angle Relationships Angles formed by one Angles formed by CHORDS:
SECANT and one TANGENT: Angles formed by CHORDS: <𝐴𝐸𝐡 B A C D A B C D E <𝐴𝐸𝐡= 𝐴𝐡 + 𝐷𝐢 2 <𝐴𝐡𝐢= 𝐴𝐷𝐡 2 Angles formed by two SECANTS: Ex 3: Ex 4: Ex 5: Angle Relationships A C B F G <𝐴𝐡𝐢 <𝐴𝐡𝐢= 𝐴𝐢 βˆ’ 𝐹𝐺 2

10 Find the π‘š<𝐷𝐸𝐴 if π‘š 𝐷𝐴 =104Β°, π‘š 𝐷𝐢 =76Β°, π‘Žπ‘›π‘‘ π‘š 𝐴𝐡 =65Β°,
Ex 3: Ex 5: B A C D A B C D E 156Β° 65Β° π‘š 𝐡𝐢 =360Β°βˆ’ 104Β°+65Β°+76Β° 104Β° π‘š 𝐡𝐢 =115Β° <𝐷𝐸𝐴= 𝐷𝐴 + 𝐡𝐢 2 π‘š 𝐡𝐷𝐴 =360Β°βˆ’156Β° 76Β° <𝐷𝐸𝐴= 104Β°+115Β° 2 π‘š 𝐡𝐷𝐴 =204Β° <𝐷𝐸𝐴=109.5Β° <𝐢𝐡𝐴= 𝐡𝐷𝐴 2 <𝐢𝐡𝐴= 204Β° 2 Ex 4: Find the π‘š<𝐴𝐡𝐢 if π‘š 𝐴𝐹 =70Β°, π‘š 𝐺𝐢 =143Β°, π‘Žπ‘›π‘‘ π‘š 𝐴𝐢 =98Β°, 70Β° A C B F G π‘š 𝐹𝐺 =360Β°βˆ’ 70Β°+98Β°+143Β° <𝐢𝐡𝐴=102Β° π‘š 𝐡𝐢 =49Β° 98Β° <𝐴𝐡𝐢= 𝐴𝐢 βˆ’ 𝐹𝐺 2 143Β° <𝐴𝐡𝐢= 98Β°βˆ’49Β° 2 <𝐴𝐡𝐢=24.5Β°

11 Day 3-4

12 Segment Relationships
One Secant and one Tangent Intersect! Two Chords Intersect! A B C D E B A C D 𝐴𝐸 βˆ— 𝐸𝐢 = 𝐷𝐸 βˆ— 𝐸𝐡 𝐴𝐷 βˆ— 𝐡𝐷 = ( 𝐢𝐷 ) 2 Segment Relationships Two Secants Intersect! EX 6: EX 7: EX 8: A C B F G 𝐴𝐡 βˆ— 𝐹𝐡 = 𝐢𝐡 βˆ— 𝐺𝐡

13 EX 6: Find 𝐴𝐸 if 𝐸𝐡 =8, 𝐷𝐸 =6, and 𝐸𝐢 =12
C B F G A B C D E 𝐴𝐸 βˆ— 𝐸𝐢 = 𝐷𝐸 βˆ— 𝐸𝐡 𝐴𝐸 βˆ—12=6βˆ—8 𝐴𝐸 βˆ—12=48 EX 8: Find 𝐴𝐡 if 𝐢𝐡 =10, 𝐺𝐡 =5, and 𝐹𝐡 =4 𝐴𝐸 =4 𝐴𝐡 βˆ— 𝐹𝐡 = 𝐢𝐡 βˆ— 𝐺𝐡 EX 7: Find 𝐢𝐷 if 𝐴𝐡 =5, 𝐡𝐷 =4 𝐴𝐡 βˆ—4=10βˆ—5 𝐴𝐷 βˆ— 𝐡𝐷 = ( 𝐢𝐷 ) 2 B A C D 𝐴𝐡 βˆ—4=50 9βˆ—4= ( 𝐢𝐷 ) 2 𝐴𝐡 =12.5 36= ( 𝐢𝐷 ) 2 6= 𝐢𝐷

14 If two chords are congruent, then the arcs are congruent
B C If two chords are congruent, then the arcs are congruent 𝑖𝑓 𝐴𝐡 ≅𝐢𝐡 π‘‘β„Žπ‘’π‘› 𝐴𝐡 β‰… 𝐢𝐡 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 π‘π‘’π‘Ÿπ‘π‘’π‘›π‘–π‘π‘’π‘™π‘Žπ‘Ÿ π‘‘π‘œ 𝐡𝐢 π‘‘β„Žπ‘’π‘› 𝐡𝐷 β‰… 𝐢𝐷 and 𝐡𝐸 β‰… 𝐸𝐢 A C B D E F G H I Two chords are congruent, if and only if they are equidistant from the center. 𝐺𝐻 β‰… 𝐡𝐢 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 𝐴𝐸 β‰… 𝐴𝐸

15 A line is tangent to a circle if and only if
it is perpendicular to a radius drawn at the point of tangency 𝐢𝐴 𝑖𝑠 π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ π‘‘π‘œ π‘π‘–π‘Ÿπ‘π‘™π‘’ 𝑇. 𝐴 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘π‘¦ A C T Z X Y If two segments from the same exterior point are tangent to a circle, then they are congruent π‘Œπ‘ π‘Žπ‘›π‘‘ π‘Œπ‘‹ π‘Žπ‘Ÿπ‘’ π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘π‘–π‘Ÿπ‘π‘™π‘’. π‘‡β„Žπ‘’π‘ . π‘Œπ‘ β‰… π‘Œπ‘‹

16 Day 5

17 Arc Length Sector Area A piece of the circumference of a circle
A slice of the circle bounded by 2 radii and an arc π΄π‘Ÿπ‘ πΏπ‘’π‘›π‘”π‘‘β„Ž= π‘₯Β° 360Β° βˆ—2πœ‹π‘Ÿ π‘†π‘’π‘π‘‘π‘œπ‘Ÿ π΄π‘Ÿπ‘’π‘Ž= π‘₯Β° 360Β° βˆ—πœ‹ π‘Ÿ 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 =150Β° E F S A R M

18 Ex 9: 𝐹𝑖𝑛𝑑 π‘‘β„Žπ‘’ π‘™π‘’π‘›π‘”π‘‘β„Ž π‘œπ‘“ π‘Žπ‘Ÿπ‘ 𝐸𝐹
𝑖𝑛 π‘π‘–π‘Ÿπ‘π‘™π‘’ 𝑆 𝑖𝑓 𝐸𝑆=6𝑖𝑛, π‘Žπ‘›π‘‘ π‘š<𝐸𝑆𝐹=60Β° π΄π‘Ÿπ‘ πΏπ‘’π‘›π‘”π‘‘β„Ž= π‘₯Β° 360Β° βˆ—2πœ‹π‘Ÿ E F S 𝐸𝐹 = 60Β° 360Β° βˆ—2πœ‹βˆ—6 𝐸𝐹 =6.3 𝑖𝑛 Ex 10: Find the area of the blue sector in Cricle M if AM=5cm and 𝐴𝑅 =150Β° π‘†π‘’π‘π‘‘π‘œπ‘Ÿ π΄π‘Ÿπ‘’π‘Ž= π‘₯Β° 360Β° βˆ—πœ‹ π‘Ÿ 2 A R M π‘†π‘’π‘π‘‘π‘œπ‘Ÿ π΄π‘Ÿπ‘’π‘Ž= 150Β° 360Β° βˆ—πœ‹ 5 2 π‘†π‘’π‘π‘‘π‘œπ‘Ÿ π΄π‘Ÿπ‘’π‘Ž=32.7 π‘π‘š 2

19 Day 6

20 Radian Measure We indicate radian measure using the symbol for a central angle 𝜽 𝑻𝒉𝒆 π’ˆπ’“π’†π’†π’Œ 𝒍𝒆𝒕𝒕𝒆𝒓 𝑻𝒉𝒆𝒕𝒂 Radian measure is the ratio of the arc length, l, to the radius of the circle 𝜽= 𝒍 𝒓 πŸ‘πŸ”πŸŽΒ°=πŸπ… π’“π’‚π’…π’Šπ’‚π’π’”

21 So if πŸ‘πŸ”πŸŽΒ°=πŸπ… π’“π’‚π’…π’Šπ’‚π’π’”, then what would πŸπŸ–πŸŽΒ°π’ƒπ’† π’Šπ’ π’“π’‚π’…π’Šπ’‚π’π’”?
πŸπŸ–πŸŽΒ°=𝝅 π’“π’‚π’…π’Šπ’‚π’π’” Lets see if we can come up with more radian measures 𝝅 𝟐 𝝅 𝟐 90Β°= ___________π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  𝝅 πŸπ… πŸ‘π… 𝟐 270Β°= ___________π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  πŸ‘π… 𝟐

22 Ex 1:

23 You Try!

24 You Try!

25 Equation of a Circle β€œCenter-Radius Form”
Any circle can be expressed in the standard form (π‘₯βˆ’β„Ž) 2 + (π‘¦βˆ’π‘˜) 2 = 𝒓 2 Where the center of the circle is at the point β„Ž,π‘˜ And the radius of the circle is 𝒓

26 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

27 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

28 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

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

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