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- 2 I understand
- 3 H2O (Water)
- 4 Apple Pie
- 5 Copyright COPI COPPY COPY
- 6 Summary mary + mary
- Prepared by: MIKEE C. TOLENTINO
- 8 identify and illustrate parts of a circle; and apply theorems in solving problems in tangent, secant segments, arcs, chords and angles of a circle.
- CIRCLE 9 It is the set of all points on a plane at a given distance from a fixed point called the center.
- 10 A circle is named by its center. Example: ʘA A center
- 11
- RADIUS 12 It is any segment joining the center to a point on the circle. radius A B Example: AB
- 13 Interior of a circle It is the set of all points in the plane of the circle whose distance from the center is less than the radius. A C Example: C
- 14 Exterior of a circle It is the set of all points outside the plane of the circle whose distance from the center is greater than the radius. A D Example: D
- 15 Congruent Circles It composed of two or more circles having congruent radii. Example: ʘA ≅ʘZ A Z
- 16 Concentric Circles It composed of two or more coplanar circles having the same center.
- CHORD 17 It is a segment joining any two points on the circle. chord E F Example: EF
- DIAMETER 18 It is a chord passing through the center. It twice the measure of the radius. diameter H G Example: GH
- 19 1. Name of the circle 2. Three radii 3. Length of radius 4. A diameter 5. A chord 6. Length of QS 7. Length of QX Let’s Try It: Q S X T R P
- ARC 20 It is a part of a circle between two points on the circle. I J Example: IJ
- 21 Semicircle It is one half of a circle and measures 180⁰. A diameter divides the circle into two semicircles. K L M N Example: NKL and LMN
- 22 Major Arc It is an arc of a circle having a measure greater than 180°. Example: OPQ O P Q
- 23 Minor Arc It is an arc smaller than a semicircle. It measures less than 180°. Example: QO O P Q
- 24 Arc Addition Postulate The measure of an arc formed by two adjacent non- overlapping arcs is the sum of the measures of those arcs.
- 25 Example 1: If VR = 54⁰;RS = 43⁰ What is VRS? Solution: mVR+mRS = mVRS 54⁰ + 43⁰ = 97⁰ R S U V
- 26 Example 2: If RSU =176⁰;RS = 49⁰ What is mSU? R S U V
- 27 Intercepted Arc The arc that lies in the interior of an angle and has endpoints on the angle. Y intercepted arc
- ANGLE 28 Central Angle An angle whose vertex is the center of the circle and whose sides are radii of the circle.
- 29 Example: Central angle and its intercepted arc are same in measure. Thus, m WAX = mWX central angle intercepted arc A W X
- Theorems on Arcs, Chords, and Central Angles 30
- Theorem 1: In the same circle, or congruent circles, two minor arcs are congruent if and only if their central angles are congruent. If, XAY≅ YAZ Then, XY≅ YZ 31 A X Y Z
- Theorem 2: In the same circle or congruent circles, two minor arcs are congruent if and only if their chords are congruent. 32 A X Y Z
- Theorem 3: If a diameter is perpendicular to a chord, then it bisects the chord and its arc. 33 A X Y Z W
- Theorem 4: In the same circle or congruent circles, two chords are equidistant from the center(s) if and only if they are congruent. 34 A X Y Z W U V
- Theorem 5: If two chords of a circle are unequal in length, then the longer chord is nearer to the to the center of the circle. 35 A X Y Z W U V
- Theorem 6: If two chords of a circle are not equidistant from the center, then the longer chord is nearer to the center of the circle. 36 A X Y Z W U V
- ANGLE 37 Inscribed Angle An angle whose vertex is on a circle and whose sides contain chords of the circle.
- 38 Inscribed Angle Theorem The measure of an angle is half the measure of its intercepted arc. inscribed angle intercepted arc Y W X
- 39 Solution: ½ WX= WXY 2 (½ WX= 45⁰ ) 2 WX = 90⁰ Y W X 45⁰ Example: What is WX?
- Corollary 1: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. 40 C D B A E
- C D B A Corollary 2: If a quadrilateral is inscribed in a circle , then its opposite angles are supplementary. 41 E
- Corollary 3: If an inscribed angle intercepts a semi-circle, then the angle is a right angle. 42 C D B A E
- Corollary 4: If two arcs of a circle are included between parallel segments, then the arcs are congruent. 43 E C D B A
- SECANT 44 It is any line, ray, or segment that intersects circle in two points. J I Example: IJ
- TANGENT 45 It is a line that intersects at exactly one point on a circle. The point is called the point of tangency.
- 46 Example: BC is a tangent BC is a tangent ray C is the point of tangency BC is a tangent segment B C
- Theorem 1: If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency. Theorem 2: If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is the tangent to the circle. 47 B C
- Corollary 1: Two tangent segments from a common external point are congruent. 48 A C B D
- Corollary 2: The two tangent rays from a common external point determine an angle that is bisected by the ray from the external point to the center of the circle. 49 A C B D
- Theorems on Tangent, Secants and Angle 50
- Theorem 1: If two chords intersect within a circle, then the measure of the angle formed is equal to one-half the sum of the measures of the intercepted arcs. If BE = 38⁰; CD = 76⁰ CAD = ½ (38⁰ +76⁰) = 57⁰ 51 B A C D E
- Theorem 2: If a tangent and a chord intersect in a point on the circle, then the measure of the angle they form is one-half the measure of the intercepted arc. If BA = 84⁰ BCA = ½ (84⁰) = 42⁰ 52 B A C D
- Theorem 3: If a tangent and a secant, two secants , or two tangents intersect in a point in the exterior of a circle, then the measure of the angle formed is equal to one-half the difference of the measures of the intercepted arcs. If BE = 70⁰; BCA=106⁰ BDA = ½ (106⁰ - 70⁰) = 18⁰ 53 B A C D E
- Theorems on Circles and Segment Lengths 54
- Theorem 1: If two chords intersect inside a circle, then the product of the lengths of the segments of one chords is equal to the product of the lengths of the segments of the other chord. 55
- Example: Find the value of AE. If BA= 4, AC = 6 and DA = 3. BA(AC)= DA(AE) (4)(6) = (3)(AE) 24 = 3AE 8 = AE 56 B A C D E
- Theorem 2: If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment is equal to the square of the length of the tangent segment. 57
- Example: Find the value of DA. If BA= 9, CA = 16. (𝐷𝐴)2 = CA(BA) (𝐷𝐴)2 = (16) (9) (𝐷𝐴)2 = 144 DA = 12 58 B A C D
- 59
- 60 1. Radii of the same circle are equal in measure. 2. Every chord is a diameter. 3. Central angle is twice the measure of its intercepted arc. 4. Major arc measures greater than 180 degree but less than 360 degree. 5. Inscribed angle is one-half the measure of its intercepted arc. Seatwork:
- 61 6. Diameter is thrice the measure of a radii. 7. Minor arc measures less than 90 degree. 8. A tangent passes through the center of a circle. Seatwork:
- 62 Seatwork: C x B A 80⁰ D 9. E
- 63 Seatwork: C x A 65⁰ D 10. 11. x 16⁰ 78⁰
- 64 Seatwork: C 53⁰ D 12. 13. x 4 5 x 99⁰ 62⁰
- 65 Seatwork: C D 14. 15. x 6 12 x 8
- “ 66
- 67

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