Ch 10 goals and common core standards Ms. Helgeson Identify segments and lines related to circles. Use properties of a tangent to a circle. Use properties of arcs of circles Use properties of chords of circles Use inscribed angles to solve problems Use properties of inscribed polygons Use angles formed by tangents and chords to solve problems in geometry.

Use angles formed by lines that intersect a circle to solve problems. Find the lengths of segments of chords. Find the lengths of segments of tangents and secants. Write the equation of a circle Use the equation of a circle and its graph to solve problems CC.9-12.G.C.2 CC.9-12.G.C.5 CC.9-12.G.C.4(+) CC.9-12.A.REI.7

Topic10 Circles

Circles Circle – The set of all points in a plane that are equidistant from a given point, called the center of the circle. Radius – The distance from the center of a circle to a point on the circle. A segment whose endpoints are the center of the circle and a point on the circle. Diameter – The distance across a circle, through its center. A chord (longest chord) that passes through the center of the circle.

Chord – A segment whose endpoints are points on the circle. Secant – A line that intersects a circle in two points. Tangent – A line that intersects a circle in exactly one point

W A C N O K Name: Radius _____ Diameter ____ Chord _____ Secant _____ Tangent _____ Tangent segment __ D Y

Always, Sometimes, Never???? Explain your answer. The longest chord of a circle is a diameter. A chord of a circle can be a radius. A tangent of a circle contains a chord. A secant of a circle contains a chord. A secant of a circle contains a diameter. A chord of a circle can be a secant.

Congruent Circle- two circles that have the same radius 5cm 5cm

Tangent Circles- coplanar circles that intersect in one point Externally Tangent Circles Internally Tangent Circles One point of intersection.

Concentric circles- coplanar circles that have a common center

10.1 Arcs and Sectors Goal Use properties of arcs of circles Use properties of chords of circles

Central Angle-an angle whose vertex is the center of a circle Minor arc- part of a circle that measure less than 180° Major arc- part of the circle that measures between 180° and 360° Semicircle- an arc whose endpoints are the endpoints of a diameter of the circle. Central angle A Minor arc Major arc P C B

You measure a minor arc by the number of degrees in its central angle You measure a minor arc by the number of degrees in its central angle. If central angle BOA has a measure of 40, then the measure of the intercepted arc AB is 40. The measure of the major are is 360 minus the measure of the minor arc making up the remainder of the circle. A 40º B 40º O C

Find the measure of each arc of R a. MN B. MPN C.PMN P N 80° M

Postulate 26: Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC B A C

Example Find the measure of each arc. CD CDB BCD B C A 148° D

Finding Measures of Arcs Find the measure of each arc.\ a. GE b.GEF c.GF G H 40° 80° 110° E F

Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? 2. A 1. Z X D 45° 45° 65° B Y C W Are the measures of the arcs congruent? Are the lengths of the arcs equal in length?

Identifying Congruent Arcs Find the measures of the blue arcs. Are they congruent? R 80° P Q 80° R S

How do you find the length s of an arc measured in degrees How do you find the length s of an arc measured in degrees? The measure of an arc is a fraction of 360˚. The arc length is a fraction of the circumference. n˚ Arc length arc measure Circumference 360˚ s n 2 r 360˚ = r = S = n (2 r) 360˚ s is the length of the arc. n is the measure of the arc.

How do you find the length s of an arc measured in radians How do you find the length s of an arc measured in radians? A radian is equal to the measure of a central angle that intercepts an arc with length equal to the radius of the circle. 1 revolution = 360˚ = 2 radians ½ revolution = 180˚ = radians

To find the arc length, use the following proportion To find the arc length, use the following proportion. arc length arc measure (radians) circumference 2 s Θ 2 r 2 s = Θ (2 r) = Θr 2 s = Θr = = The variable theta Θ is often used for angles measured in radians.

Converting to radian measure: radian meas. = (deg. Measure) · (π/180) 60˚ = (60) · (π/180) = π/3 120˚ = (120) · (π/180) = (2/3) π Converting to degree measure: Deg. Meas. = (radian meas.)(180/π) π/4 = (π/4)(180/π) = 45˚

Area = πr² Area of a sector = (m/360) · πr² Area of a sector = (m/2π) · πr² = (m/2)r²

In a circle with radius 4, what is the length of an arc that has a measure of 80˚? Round to the nearest tenth. In a circle with radius 6, what is the length of an arc that has a measure of radians? Round to the nearest tenth. In a circle with radius 10, what is the area of a sector that has a measure of 8 radians? Round to the nearest tenth. Radius = 3, angle measure 165˚. Area of S?

A Segment of a Circle p 422 A segment of a circle is the part of a circle bounded by an arc and the segment joining its endpoints. What is the area of the shaded region? Area of segment = area of sector – area of ∆ Trig: A = ½ r² ( - sin )

What is the area of each segment? Central angle = 90˚, r = 10. Measure of arc = 144˚, r = 5. Page 423, example 6

10.2 Lines Tangent to a Circle A tangent to a circle is a line, segment, or ray in the plane of the circle that intersects the circle in exactly one point. That point is the point of tangency.

Common tangent- a line segment that is tangent to two coplanar circles. Common internal tangent- intersects the segment that joins the centers of the two circles. Common external tangent- does not intersect the segment that joins the centers of the two circles. Common external tangent Common Internal tangent D A C B

Draw two circles having no common tangents. Draw two circles having one common external tangent. Draw two circles having two common external tangents and no common internal tangents. Draw two circles having two common external tangents and one common internal tangent. Draw two circles having two common external tangents and two common internal tangents Draw an example with exactly 4 common tangents.

Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. P Q l

Theorem - In a plane, if the line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. P Q l

Examples IS CE tangent to D? Explain. E 43 C 11 45 D

You are standing 14 ft. From a water tower You are standing 14 ft. From a water tower. The distance from you to a point of tangency on the tower is 28 ft. What is the radius of the water tower? B 28 ft C r 14 ft r A

BC is tangent to A. Find the radius of the circle. 16 r C B 24

Ex: Line m is tangent to circle T at B, and line n is tangent to circle T at C. What is the value of x? m B 135˚ T x˚ A C n

Theorem If two segments from the same exterior point are tangent to a circle, then they are congruent. R T P S

AB is tangent to C at B. AD is tangent to C at D. Find the value of x. 21 D

P 431 Construct Tangent Lines

10.3 Chords A chord is a segment whose endpoints are on a circle.

Theorem: If two chords in a circle or in congruent circles are congruent, then their central angles are congruent. Converse: If two central angles in a circle or in congruent circles are congruent, then their chords are congruent.

Theorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AB = BC if and only if AB = BC ~ ~ A C B

Find mAD using Theorem 10.4 (x + 40)° D 2x° C B A

Use Theorem 10.4 to find mBC B (2x + 48)° (3x + 11)° A C D

Theorem 10.6 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. DE = EF, DG =GF ~ ~ F E G D

Converse of thm 10.6 If a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord.

Theorem 10.7 The perpendicular bisector of a chord contains the center of the circle.

Theorem If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. JK is a diameter of the circle M J K L

Theorem 10.7 In the same circle, or in congruent circles two chords are congruent if and only if they are equidistant from the center. AB = CD if and only if EF = EG ~ ~ C G E D A B F

Using Theorem 10.7 AD = 40, CD =25. Find CG. A B C G F D E

Example AB = 8; DE = 8; CD = 5. Find CF A F C B 5 E G D

10.4 Inscribed Angles

Vocabulary: Inscribed angle- an angle whose vertex is on a circle and whose sides contain chords of the circle Intercepted arc- an arc that lies in the interior of an inscribed angle and has endpoints on the angle

Measure of an Inscribed Angle Theorem 10.8 Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. m ADB = ½ mAB A C B D

Find the measure of the highlighted arcs W B. Z T Q 115° X Y

Find the measure of the arc mADC B A C D

Find the measure of angle ABC m B C A 196°

Find m ACB, m ADB, and m AEB A 60° B E D C

Find the measurements of angles ABE; ACE; and ADE 90° E

Theorem 10.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. C = D A ~ C B D

Find the measure of an angle It is given that the measurement of angle E is 75°. What is the measurement of angle F? G E 75° F H

It is given that the measurement of angle B is 44° It is given that the measurement of angle B is 44°. What is the measurement of angle C? B C A D

Theorem 10.9 The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc.

m<1 = ½ mAB m<2 = ½ mBCA

Line m is tangent to the circle. Find mRST 102° S T

Line m is tangent to the circle Line m is tangent to the circle. Find the measure of the red triangle or arc. S m 130° R P

Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

B is a right angle if and only if AC is diameter of the circle

Theorem 10.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

D, E, F, and G lie on same circle, C, if and only if m<D + m<F = 180° and m<E + m<G =180°. E F C D G

Using Theorems 10.10 & 10.11 Find the value of each variable 120° z° Q 80° F y° G 2x° A C

Find the value of the variable 3x° C A

Find the value of the variable S T x° y° 85° 80° U R

Using an inscribed quadrilateral 2y° P D 3y° 3x° B 5x° C

10.5 Secant Lines and Segments

Theorem 10.10 If two chords or secant lines intersect in the interior of a circle, then the measure of the angle is one half the sum of the measures of the intercepted arcs.

m<1 = ½(mCD + mAB), m<2 = ½(mBC + mAD)

Finding the measure of an angle formed by two chords 106° S P x° Q R 174°

Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Theorem 14. continued M< = ½ (mBC - mAC) B A 1 C

Theorem 14. continued M< 2 = ½(mPQR - mPR) P 2 Q R

Circumscribed Angle: A circumscribed angle of a circle and its associated central angle are supplementary.

Theorem 14. continued m<3 = ½(mXY - mWZ) X W 3 Z Y

Using Theorem 10.14 Find the value of X T R P X 80° S

Find the value of X using Theorem 10.14 200° D F X° 72° H G

Section 5 Segment Lengths in Circles

Theorem 10.12 If two chords intersect in the interior of 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

Theorem 10.12 B C E A D EA • EB = EC • ED

Finding Segments Lengths Chords ST and PQ intersect inside the circle. Find the value of x. S 3 Q P 9 R X 6 T

Theorem 10.12 If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment

Theorem 10.12 B A E C D EA • EB = EC • ED

Using theorem 10.12 solve for X Q P 11 9 R x 10 S T

Find the value of X and Y 20 25 F E D Y 14 8 24 C 16 G X H

Theorem 10.12 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

Theorem 10.12 A E C D (EA)² = EC • ED

Using Theorem 10.12 solve for X. 24 E GC = X 12 D G x C

Example 30 B A x C 24 D