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Geometry Circles 2014 -03 -31 www. njctl. org

Table of Contents Click on a topic to go to that section Parts of a Circle Angles & Arcs Chords, Inscribed Angles & Polygons Tangents & Secants Segments & Circles Equations of a Circle Area of a Sector

Parts of a Circle Return to the table of contents

A circle is the set of all points in a plane that are a fixed distance from a given point in the plane called the center

. and is named by a capital letter placed by the center of The symbol for a circle is the circle. (circle A or . A) is a radius of A . A A radius (plural, radii) is a line segment drawn from the center of the circle to any point on the circle. It follows from the B definition of a circle that all radii circle are congruent. of a

M is a chord of circle R A A chord is a segment that has its endpoints on the circle. C is the diameter of circle A A diameter is a chord that goes through the center of the circle. All diameters of a circle are congruent. T What are the radii in this diagram? Answer A

The relationship between the diameter and the radius T The measure of the diameter, d, is twice the measure of the radius, r. A Therefore, or C Answer M In. A If , then what is the length of

A diameter of a circle is the longest chord of the circle. True False Answer 1

A radius of a circle is a chord of a circle. True False Answer 2

Two radii of a circle always equal the length of a diameter of a circle. True False Answer 3

If the radius of a circle measures 3. 8 meters, what is the measure of the diameter? Answer 4

How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many Answer 5

A secant of a circle is a line that intersects the circle at two points. A line l is a secant of this circle. B l D E k A tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency). line k is a tangent D is the point of tangency. tangent ray, , and the tangent segment, , are also called tangents. They must be part of a tangent line. Note: This is not a tangent ray.

COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. . . 2 points . . . tangent circles 1 point concentric circles no points

A Common Tangent is a line, ray, or segment that is tangent to 2 coplanar circles. Internally tangent (tangent line passes between them) Externally tangent (tangent line does not pass between them)

How many common tangent lines do the circles have? Answer 6

How many common tangent lines do the circles have? Answer 7

How many common tangent lines do the circles have? Answer 8

How many common tangent lines do the circles have? Answer 9

Using the diagram below, match the notation with the term that best describes it: . B . A C . . D . F . Answer E . G center secant diameter common tangent chord radius tangent point of tangency

Angles & Arcs Return to the table of contents

An ARC is an unbroken piece of a circle with endpoints on the circle. . A Arc of the circle or . AB B Arcs are measured in two ways: 1) As the measure of the central angle in degrees 2) As the length of the arc itself in linear units (Recall that the measure of the whole circle is 360 o. )

A central angle is an angle whose vertex is the center of the circle. . . H In T , is the central angle. A Name another central angle. Answer S M

If is less than 1800, then the points on that lie in the interior of form the minor arc with endpoints M and H. . . minor arc MA Answer S M H Highlight T A Name another minor arc. MA

major arc . . H T Answer S M A Points M and A and all points of exterior to form a major arc, Major arcs are the "long way" around the circle. MSA Major arcs are greater than 180 o. Highlight Major arcs are named by their endpoints and a point on the arc. Name another major arc. MSA

. . minor arc H T Answer S M A A semicircle is an arc whose endpoints are the endpoints of the diameter. MAT is a semicircle. Highlight the semicircle. Semicircles are named by their endpoints and a point on arc. Name another semicircle. the

Measurement By A Central Angle The measure of a minor arc is the measure of its central angle. The measure of the major arc is 3600 minus the measure of the central angle. B 400 . D A 400 G 3600 - 400 = 3200

The Length of the Arc Itself (AKA - Arc Length) Arc length is a portion of the circumference of a circle. Arc Length Corollary - In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 3600. C A arc length of CT = CT 3600 or r T arc length of CT = CT 3600 .

EXAMPLE In A, the central angle is 600 and the radius is 8 cm. Find the length of CT Answer C 8 cm A 600 T

EXAMPLE A, the central angle is 400 and the length of In Find the circumference of SY is 4. 19 in. A. S 4. 19 in 400 Answer A Y

In circle C where is a diameter, find B 1350 D C 15 in A Answer 10

In circle C, where is a diameter, find B 1350 D C 15 in A Answer 11

In circle C, where is a diameter, find B 1350 D C 15 in A Answer 12

In circle C can it be assumed that AB is a diameter? B Yes No 1350 D C A Answer 13

Find the length of A B 450 3 cm C Answer 14

Find the circumference of circle T. T 750 6. 82 cm Answer 15

16 In circle T, WY & XZ are diameters. WY = XZ = 6. If XY = A , what is the length of YZ? 1400 X W C Answer B T D Z Y

ADJACENT ARCS Adjacent arcs: two arcs of the same circle are adjacent if they have a common endpoint. Just as with adjacent angles, measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs. . C T . . A = +

EXAMPLE A result of a survey about the ages of people in a city are shown. T measures. S 1. >65 2. 300 900 U 1000 17 -44 Answer 3. Find the indicated 800 4. 600 45 -64 R 15 -17 V

Match the type of arc and it's measure to the given arcs below: T Q 1200 600 R S minor arc 800 major arc 1200 1600 semicircle 1800 2400 Teacher Notes 800

CONGRUENT CIRCLES & ARCS Two circles are congruent if they have the same radius. Two arcs are congruent if they have the same measure and they are arcs of the same circle or congruent circles. T D C R E 550 S U F because they are in the same circle and & have the same measure, but are not congruent because they are arcs of circles that are not congruent.

A 17 B True 700 1800 400 False C Answer D

18 M L True 850 P N Answer False

19 Circle P has a radius of 3 and length ? 900 of has a measure of . What is the A A B P C D Answer B

Two concentric circles always have congruent radii. True False Answer 20

If two circles have the same center, they are congruent. True False Answer 21

Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece? Answer 22

Chords, Inscribed Angles & Polygons Return to the table of contents

When a minor arc and a chord have the same endpoints, we call the arc The Arc of the Chord. P . C **Recall the definition of a chord - a segment with endpoints on the circle. is the arc of Q

THEOREM: In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. T is the perpendicular bisector of . S Therefore, is a diameter of the circle. Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle. E Q P

THEOREM: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. A C is a diameter of the circle and is perpendicular to chord . Therefore, X S E

THEOREM: In the same circle, or in congruent circles, two minor arcs are congruent if and only if th corresponding chords are congruent. B C iff A D *iff stands for "if and only if"

BISECTING ARCS X If , then point Y and any line segment, or ray, that contains Y, bisects C Y Z

EXAMPLE Find: C B , . , and (9 x)0 D E (80 - x)0 Answer A

THEOREM: In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center. C . G D E A F B iff

EXAMPLE Given circle C, QR = ST = 16. Find CU. Since the chords QR & ST are congruent, they are equidistant from C. Therefore, R Q . 2 x C 5 x - 9 V S T Answer U

In circle R, and . Find A 1080 C . R Answer 23 B D 1

Given circle C below, the length of A 5 B 10 C 15 D is: A D B 10 . 20 F Answer 24 C

Given: circle P, PV = PW, QR = 2 x + 6, and ST = 3 x - 1. Find the length of QR. A 1 B 7 C 20 D 8 R . V Q S P W T Answer 25

AH is a diameter of the circle. A True 5 3 False M T 3 S H Answer 26

INSCRIBED ANGLES D Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle. O G The arc that lies in the interior of an inscribed angle, and has endpoints on the angle, is called the intercepted arc. is an inscribed angle and is its intercepted arc.

THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. C T A

EXAMPLE Find and . T 500 480 P S Answer R Q

THEOREM: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A since they both intercept B D C

In a circle, parallel chords intercept congruent arcs. C D O A In circle O, if . B , then

Given circle C below, find D E . C 1000 Answer 27 350 A B

Given circle C below, find D E . C 1000 350 A B Answer 28

29 Given the figure below, which pairs of angles are congruent? R A S C U D T Answer B

Find X Y . Z P Answer 30

In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 1200. Find the measure of one of the arcs included between the chords. Answer 31

Given circle O, find the value of x. x B A 300 C . O D Answer 32

Given circle O, find the value of x. 1000 B A 350 . O Answer 33 C D x

Try This and In the circle below, Find , and Q 2 1 3 4 S T Answer P

INSCRIBED POLYGONS A polygon is inscribed if all its vertices lie on a circle. . . inscribed triangle inscribed quadrilateral. . .

THEOREM: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. A . x iff AC is a diameter of the circle. L G

THEOREM: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. E . N C R N, E, A, and R lie on circle C iff A

EXAMPLE Find the value of each variable: M L 2 a K 4 b 2 a J Answer 2 b

The value of x is A 1500 B 980 C D C 680 B x 820 D 1120 y 1800 A Answer 34

In the diagram, ? A 150 B 300 C 600 D 1200 is a central angle and . What is A B D . C Answer 35

What is the value of x? A 5 B 10 C 13 E (12 x + 40)0 (8 x + 10)0 D 15 F G Answer 36

Tangents & Secants Return to the table of contents

**Recall the definition of a tangent line: A line that intersects the circle in exactly one point. THEOREM: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle the(point of tangency). Line is l tangent to circle X iff B would be the point of tangency. l X . . B l

Verify A Line is Tangent to a Circle Given: T Is is a radius of circle P tangent to circle P? 12 . 35 37 Answer } S P

Finding the Radius of a Circle If B is a point of tangency, find the radius of circle C. A 50 ft 80 ft r B Answer . C r

THEOREM: Tangent segments from a common external point are congruent. R . P A T If AR and AT are tangent segments, then

EXAMPLE Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x. S 28 . C 3 x + 4 T Answer R

37 AB is a radius of circle A. Is BC tangent to circle A? 25 . A 60 C } No B 67 Answer Yes

38 S is a point of tangency. Find the radius r of circle T. A 31. 7 B 60 T C 14 r D 3. 5 . r 36 cm 48 cm Answer S R

In circle C, DA is tangent at A and DB is tangent at B. Find x. A 25 . C D 3 x - 8 B Answer 39

AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC. B F E C Answer 40 . O A D

Tangents and secants can form other angle relationships in circle. Recall the measure of an inscribed angle is 1/2 its intercepted arc. This can be extended to any angle that has its vertex on the circle. This includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents.

A Tangent and a Chord THEOREM: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. . M A . . 2 R 1

A Tangent and a Secant, Two Tangents, and Two Secants THEOREM: If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is half the difference of its intercepted arcs. a tangent and a secant two tangents two secants B X . P Q A 1 W 2 Y 3 C M Z

THEOREM: If two chords intersect inside a circle, then the measure of each angle is half the sum o the intercepted arcs by the angle and vertical angle. A M 1 2 H T

EXAMPLE Find the value of x. D x 0 760 1780 B A Answer C

EXAMPLE Find the value of x. x 0 1560 Answer 1300

Find the value of x. C 780 H 420 E x 0 F D Answer 41

Find the value of x. (3 x - 2)0 (x + 6)0 340 Answer 42

Find B Answer 43 650 A

Find 2600 1 Answer 44

Find the value of x. . 50 x 122 450 Answer 45

To find the angle, you need the measure of both intercepted arcs. First, find the measure of the minor arc. Then we can calculate the measure of the angle. x 0 2470 x 0 A Answer B

Find the value of x. 2200 x 0 Answer 46

Find the value of x. x 0 1000 Answer 47

Find the value of x Answer 48 x 0 500

Find the value of x. 1200 (5 x + 10)0 Answer 49

Find the value of x. (2 x - 30)0 300 x Answer 50

Segments & Circles Return to the table of contents

THEOREM: If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal. C A E D B

EXAMPLE 4 5 5 x Answer Find the value of x.

EXAMPLE Find ML & JK. K x+2 L x J x + x+1 4 Answer M

Find the value of x. x 16 9 18 Answer 51

52 A Find the value of x. -2 x 4 2 x + 6 C D 5 6 2 x Answer B

THEOREM: If two secant segments are drawn to a circle from an external point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment. B A E C D

EXAMPLE Find the value of x. x 6 5 Answer 9

Find the value of x. 3 x+2 x+1 Answer 53 x-1

Find the value of x. 5 4 x+4 x-2 Answer 54

THEOREM: If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment. A E C D

EXAMPLE Find RS. Q R x S 8 T Answer 16

Find the value of x. 3 1 x Answer 55

Find the value of x. 24 12 x Answer 56

Equations of a Circle Return to the table of contents

(x, y) r Let (x, y) be any point on a circle with center at the origin and radius, r. By the Pythagorean Theorem, y x x 2 + y 2 = r 2 This is the equation of a circle with at the origin. center

EXAMPLE 4 Answer Write the equation of the circle.

For circles whose center is not at the origin, we use the distance formula to derive the equation. (x, y) . r (h, k) This is the standard equation of a circle.

EXAMPLE Answer Write the standard equation of a circle with center (-2, 3) & radius 3. 8.

EXAMPLE Answer The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.

EXAMPLE The equation of a circle is (x - 4)2 + (y + 2)2 = 36. Graph the circle. We know the center of the circle is (4, -2) and the radius is 6. . . First plot the center and move 6 places in each direction. . . Then draw the circle.

What is the standard equation of the circle below? A x 2 + y 2 = 400 B (x - 10)2 + (y - 10)2 = 400 C (x + 10)2 + (y - 10)2 = 400 D (x - 10)2 + (y + 10)2 = 400 10 Answer 57

What is the standard equation of the circle? A (x - 4)2 + (y - 3)2 = 81 B (x - 4)2 + (y - 3)2 = 9 C (x + 4)2 + (y + 3)2 = 81 D (x + 4)2 + (y + 3)2 = 9 Answer 58

What is the center of (x - 4)2 + (y - 2)2 = 64? A (0, 0) B (4, 2) C (-4, -2) D (4, -2) Answer 59

What is the radius of (x - 4)2 + (y - 2)2 = 64? Answer 60

A 2 B 4 C 8 D 16 Answer The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle? 61

Which point does not lie on the circle described by the equation (x + 2)2 + (y - 4)2 = 25? A (-2, -1) B (1, 8) C (3, 4) D (0, 5) Answer 62

Area of a Sector Return to the table of contents

A sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them. B Minor Sector Major Sector A C

63 Which arc borders the minor sector? B Answer A A C D B

64 Which arc borders the major sector? W Answer A B X Y Z

Lets think about the formula. . . The area of a circle is found by We want to find the area of part of the circle, so the formula for the area of a sector is the fraction of the circle multiplied by the area of the circle When the central angle is in degrees, the fraction of the circle is out of the total 360 degrees.

Finding the Area of a Sector 1. Use the formula: hen θ is in degrees A r=3 450 C Answer B

Example: Find the Area of the major sector. C A Answer 8 cm 600 T

Find the area of the minor sector of the circle. Round your answer to the nearest hundredth. C 5. 5 cm A T 300 Answer 65

Find the Area of the major sector for the circle. Round your answer to the nearest thousandth. C Answer 66 12 cm A 850 T

What is the central angle for the major sector of the circle? C Answer 67 15 cm A 1200 G

Find the area of the major sector. Round to the nearest hundredth. C Answer 68 15 cm A 1200 G

The sum of the major and minor sectors' areas is equal to the total area of the circle. True False Answer 69

A group of 10 students orders pizza. They order 5 12" pizzas, that contain 8 slices each. If they split the pizzas equally, how many square inches of pizza does each student get? Answer 70

You have a circular sprinkler in your yard. The sprinkler has a radius of 25 ft. How many square feet does the sprinkler water if it only rotates 120 degrees? Answer 71