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Bell work What is a circle? Bell work Answer A circle is a set of all points in a plane that are equidistant from a given point, called the center of the circle Unit 3 : Circles: 10.1 Line & Segment Relationships to Circles (Tangents to Circles) Objectives: Students will: 1. Identify Segments and lines related to circles. 2. Use Properties of a tangent to a circle Lines and Segments related to circles Exterior Point CHORD CENTER • • TANGENT LINE • DIAMETER ALSO A CHORD Interior Point SECANT RADIUS Point of tangency Lines and Segments related to circles Center of the circle CENTER •P Circle P Lines and Segments related to circles Diameter – from one point on the circle passing through the center (2 times the radius) CENTER • DIAMETER ALSO A CHORD Lines and Segments related to circles Radius – Segment from the center of the circle to a point on the circle (1/2 the diameter) CENTER • RADIUS (I) = 1/2 the Diameter Lines and Segments related to circles Chord – a segment from one point on the circle to another point on the circle CHORD • DIAMETER ALSO A CHORD Lines and Segments related to circles Secant – a line passing through two points on the circle • SECANT Lines and Segments related to circles Tangent – is a line that intersects the circle at exactly one point TANGENT LINE • • Point of Tangency Label Circle Parts 1. 2. 3. 4. Semicircles Center Diameter Radius 5. Exterior 6. Interior 7. Diameter 8. Chord 9. Tangent 10. Secant 11. Minor Arc 12. Major Arc Point of Tangency Theorem If a line is tangent to a circle, then it is perpendicular ( _|_ ) to the radius drawn to the point of tangency. Q Tangent line P k If line k is tangent to circle Q at point P, Then line k is _|_ to Segment QP. Example 1 Find the distance from Q to R, given that line m is tangent to the circle Q at Point P, PR = 4 cm and radius is 3 cm. Q • 3 cm P 4 cm •R m Example 1 answer Use the Pythagorean Theorem a² + b² = c² 3² + 4² = c² 9 + 16 = c² √25 = √c² c =5 Example 2 Given that the radius (r) = 9 in, PR = 12, and QR = 16 in. Is the line m tangent to the circle? Q 9 in • P 12 in 16 in •R m Example 2 answer No, it is not tangent. Use the Pythagorean Theorem a² + b² = c² 9² + 12² = 16² 81 + 144 = 256 225 = 256 Since they are not = then the triangle is not a right triangle and thus the radius is not perpendicular to the line m, therefore the line is not tangent to the circle. Intersections of Circles No Points of Intersection CONCENTRIC CIRCLES – Coplanar circles that share a common center point • Intersections of Circles One Point of Intersection The Circles are tangent to each other at the point • Internal Tangent External Tangent Common Tangents • Intersections of Circles Two Points of Intersection • • Tangents drawn from a point not on the circle theorem If two segments from the same exterior point are tangent to a circle, then they are congruent. R • • •P • T S __ __ If SR and ST are tangent to circle P, __ __ SR ST Example 3 Segment SR and Segment ST are tangent to circle P at Points R and T. Find the value of x. 2x + 4 R • • •P • T 3x – 9 S Example 3 Answer __ __ Since SR and ST are tangent to the circle, then the segments are , so 2x + 4 = 3x – 9 -2x -2x 4=x–9 +9 +9 13 = x Unit 3 : Circles: 10.2 Arcs and Chords Objectives: Students will: 1. Use properties of arcs and chords to solve problems related to circles. Bell work Find the value of radius, x, if the diameter of a circle is 25 ft. 25 ft x Arcs of Circles CENTRAL ANGLE – An angle with its vertex at the center of the circle Central Angle • CENTER P P • A 60º • B Arcs of Circles Minor Arc AB and Major Arc ACB Central Angle • CENTER P P • MAJOR ARC ACB C • A MINOR ARC AB 60º • B Arcs of Circles The measure of the Minor Arc AB = the measure of the Central Angle The measure of the Major Arc ACB = 360º - the measure of the Central Angle Central Angle • CENTER P P The measure of the MAJOR ARC = 360 – the measure of C the MINOR ARC ACB = 360º - 60º = 300º • • 300º A 60º • B Measure of the MINOR ARC = the measure of the Central Angle AB = 60º Label Circle Parts 1. 2. 3. 4. Semicircles Center Diameter Radius 5. Exterior 6. Interior 7. Diameter 8. Chord 9. Tangent 10. Secant 11. Minor Arc 12. Major Arc Arcs of Circles Semicircle – an arc whose endpoints are also the endpoints of the diameter of the circle; Semicircle = 180º 180º • Semicircle Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs A • 170º + 8 0º = 2 5 0º ARC ABC = 250º • 170º • AB + BC = ABC 80º •B C Example 1 Find m XYZ and XZ X • • 75º P• 110° • Z Y Congruent Chords Theorem In the same circle or in congruent circles two minor arcs are congruent iff their corresponding chords are congruent Congruent Arcs and Chords Theorem Example 1: Given that Chords DE is congruent to Chord FG. Find the value Arc DE = 100º of x. D E F G Arc FG = (3x +4)º Congruent Arcs and Chords Theorem Example 2: Given that Arc DE is congruent to Arc FG. Find the value of x. E D Chord DE = 25 in Chord FG = (3x + 4) in F G Diameter through a chord If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chords and its arcs. Chord Diameter Congruent Arcs P • Congruent Segments Diameter through a chord If one chord is the perpendicular bisector of another chord then the first chord is the diameter Chord 2 Chord 1: _|_ bisector of Chord 2, Chord 1 = the diameter P • Diameter Distance Congruent Chords are from the Center In the same circle or in congruent circles, two chords are congruent iff they are equidistant from the center. (Equidistant means same perpendicular distance) Q T Chord TS Chord QR __ __ iff PU VU V • U P S R Center P Example Find the value of Chord QR, if TS = 20 inches and PV = PU = 8 inches Q T V 8 in • U P 8 in S R Center P Unit 3 : Circles: 10.3 Arcs and Chords Objectives: Students will: 1. Use inscribed angles and properties of inscribed angles to solve problems related to circles Bell work 1 Find the measure of Arc ABC, if Arc AB = 3x, Arc BC = (x + 80º), and __ __ AB BC A C AB = 3xº BC = ( x + 80º ) B Bell work 2 You are standing at point X. Point X is 10 feet from the center of the circular water tank and 8 feet from point Y. Segment XY is tangent to the circle P at point Y. What is the radius, r, of the circular water tank? Y r P 8 ft 10 ft • • X Words for Circles 1. 2. 3. 4. Inscribed Angle Intercepted Arc Inscribed Polygons Circumscribed Circles Are there any words/terms that you are unsure of? Inscribed Angles Inscribed angle – is an angle whose vertex is on the circle and whose sides contain chords of the circle. A INSCRIBED ANGLE INTERCEPTED ARC, AB B Vertex on the circle Intercepted Arc Intercepted Arc – is the arc that lies in the interior of the inscribed angle and has endpoints on the angle. A INSCRIBED ANGLE INTERCEPTED ARC, AB B Vertex on the circle Inscribed Angle Theorem. The measure of an inscribed angle is equal half of the measure of its intercept arc. Central Angle • CENTER P B = ½ m AC P • • Inscribed angle A m ∕_ ABC C Example 1 The measure of the inscribed angle ABC = ½ the measure of the intercepted AC. Central Angle • B m ∕_ ABC = ½ mAC = 30º • 30º • A 60º •C Measure of the INTERCEPTED ARC = the measure of the Central Angle AC = 60º Example 2 Find the measure of the intercepted TU, if the inscribed angle R is a right angle. T • R • • U Example 3 Find the measure of the inscribed angles Q , R ,and S, given that their common intercepted TU = 86º Q • R T • S •U TU = 86º Inscribed Angles Intercepting the Same Arc Theorem If two inscribed angles of a circle intercepted the same arc, then the angles are congruent Q • IF ∕_ Q and ∕_ S both intercepted TU, then T • ∕_ Q ∕_ S S •U Inscribed vs. Circumscribed Inscribed polygon – is when all of its vertices lie on the circle and the polygon is inside the circle. The Circle then is circumscribed about the polygon Circumscribed circle – lies on the outside of the inscribed polygon intersecting all the vertices of the polygon. Inscribed vs. Circumscribed The Circle is circumscribed about the polygon. Circumscribed Circle Inscribed Polygon Inscribed Right Triangle Theorem If a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Hypotenuse = Diameter • Converse 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 a right angle. The triangle is inscribed in the circle and one of its sides is the diameter Angle B is a right angle and measures 90º • Diameter = Hypotenuse B Example Triangle ABC is inscribed in the circle Segment AC = the diameter of the circle. Angle B = 3x. Find the value of x. A • C 3xº B Inscribed Quadrilateral Theorem A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary. X • • Y / X + / Z = 180º, and • P W • The Quadrilateral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Example A quadrilateral WXYZ is inscribed in circle P, if ∕_ X = 103º and ∕_ Y = 115º , Find the measures of ∕_ W = ? and ∕_ Z = ? X • • 103º Y 115º / X + / Z = 180º, and • P W • The Quadrilateral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Example From Theorem 10.11 ∕_ W = 180º – 115º = 65º and ∕_ Z = 180º – 103º = 77º X • • 103º Y 115º / X + / Z = 180º, and • P W • The Quadrilteral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Unit 3 : Circles: 10.4 Other Angle Relationships in Circles Objectives: Students will: 1. Use angles formed by tangents and chords to solve problems related to circles 2. Use angles formed by lines intersecting on the interior or exterior of a circle to solve problems related to circles Bell work 1 Find the measure of the inscribed angles , R, given that their common intercepted TU = 92º T • TU = 92º R • • U Bell work 2 A quadrilateral WXYZ is inscribed in circle P, if ∕_ X = 130º and ∕_ Y = 106º , Find the measures of ∕_ W = ? and ∕_ Z = ? X • • 130º Y 106º / X + / Z = 180º, and • P W • The Quadrilateral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Angle Formed by 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 ½ the measure of its intercepted arc A • m ∕_ 2 B • = ½ m Major ABC Angle 2 P Angle 1 • • C m m ∕_ 1 = ½ m minor AC Example 1 Find the measure of Angle 1 and Angle 2, if the measure of the minor Arc AC is 130º A • B • P • Angle 1 Angle 2 • C m m minor AC = 130º Example 2 Find the measure of Angle 1, if Angle 1 = 6xº, and the measure of the minor Arc AC is (10x + 16)º m minor AC = (10x + 16)º A • P • B Angle 1= 6xº • • m C Intersections of lines with respect to a circle There are three places two lines can intersect with respect to a circle. • • • On the circle In the circle Outside the cirlce Angles Formed by the Intersection of Two Chords Theorem If two chords intersect in the interior of a circle , then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle. D m ∕_ 1 Angle 1 • = ½ (m AB + m CD) • P C • Angle 2 A • • B m ∕_ 2 = ½ (mBC + mAD) Example Find the value of x. m CD = 16º D • • P A m AB = 40º • C • Angle 1 xº • B Angles formed by secants and tangents: 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 ½ the difference of the intercepted arcs. 1 Tangent and 1 Secant 2 Tangents P B • A • 1 • C 2 Secants X • • 2 • • R • • W •Q • m ∕_ 1 m ∕_ 2 = ½ (m BC – m AC) B = ½ (m PQR – m PR) 3 • • Z • Y m ∕_ 3 = ½ (m XY – m WZ) Example 1 Find the value of x P • • Major Arc PQR = 266º xº •Q • R m ∕_ x = ½ (m PQR - mPR) Example 2 Find the value of x, GF. The m EDG = 210º The m angle EHG = 68º E • Major Arc EDG = 210º F • D • xº 68º • H • m ∕_ EHG = 68º = ½ (m EDG – m GF) G Unit 3 : Circles: 10.5 Segment Lengths in Circles Objectives: Students will: 1. Find lengths of segments of chords, secants and tangents Bell work 1 Find the value of x. m CD = 20º D • • P A m AB = 65º • C • Angle 1 xº • B Bell work 2 Find the value of x, GF. The m EDG = 300º The m angle EHG = 54º E • Major Arc EDG = 200º F • D • xº 54º • H • m ∕_ EHG = 54º = ½ (m EDG – m GF) G Answer m ∕_ EHG = 68º = ½ (m EDG – m GF) 54º = ½ ( 200º - xº ) 108º = 200º - xº E xº = 200º - 108º • xº = 92º Major Arc EDG = 200º F • D • xº 54º • H • m ∕_ EHG = 54º = ½ (m EDG – m GF) G Bell work 3 A quadrilateral WXYZ is inscribed in circle P, if ∕_ X = 130º and ∕_ Y = 106º , Find the measures of ∕_ W = ? and ∕_ Z = ? X • • 130º Y 106º / X + / Z = 180º, and • P W • The Quadrilateral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Bellwork 3 Answer From Theorem 10.11 ∕_ W = 180º – 106º = 74º and ∕_ Z = 180º – 130º = 50º X • • 130º Y 106º / X + / Z = 180º, and • P W • The Quadrilteral WXYZ is inscribed in the circle iff • / W + / Y = 180º Z Segments of Chords, Secants and Tangents External Segment Segments of Chords Segments of Secants Segments of Tangents Lengths of Intersecting Chords Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord A • EA • EB = EC • ED C • E D • B Example Chord AB intersects Chord CD in the interior of the circle at Point E, Find the measure of segment EA. Given ED= 6cm, EC = 16 cm, and EB = 8 cm. A • EA • EB = EC • ED 16 cm C • ? 6 cm E D 8 cm • B Example Answer EA = 12 cm A • EA • EB = EC • ED EA • 8 = 16 • 6 8EA = 96 8 8 EA = 12 cm 12 cm 16 cm C • 6 cm E D 8 cm • B Lengths of Intersecting Secants If two secant segments Theorem share the same endpoint outside the circle, then the product of the length of one secant segment and the length of the of its external segment equals the product of the length of the other secant segment and the length of its external segment. B A EA • EB = EC • ED E C D Example EA = 9 ft, AB = 11 ft, and EC = 10 ft. Find CD, the value of x and ED. B 9 ft EA • EB = EC • ED A 11 ft E 10 ft C x D Example Answer CD = x = 8 ft ED = 18 ft EA • EB = EC • ED 9 (9+11) = 10(10 + x) 9 (20) = 100 + 10x 180 = 100 + 10x -100 -100 E 80 = 10x 10 10 x = 8 ft and ED = (10+8) ED = 18 ft B 9 ft 10 ft A C 11 ft x D Lengths of Intersecting Tangent and Secants Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the square of the length of the tangent segment A (EA)² = EC • ED E C D Example EC = 9 in, CD = 15 in Find EA, the value of x. A x (EA)² = EC • ED E 9 C 15 in D Example Answer Find EC, the value of x = 5, and ED= 20 (EA)² = EC • ED A 10² = x • (x + 15) 10 in 100 = x² + 15x 0 = x² + 15x - 100 x = -15 ± √15²- 4(1)(-100) 2 x = -15 + √625 = -15 +25 2 2 x = (10/2) = 5 E x C 15 in D 10.6 Equations of Circles TSW write the standard equation of a circle given the radius and center TSW determine the radius and center of a circle given the equation in standard form Standard form of a Circle (x - h)2 + (y – k)2 = r2 Center of Circle: ( h, k) Radius: r Example 1 Find the radius and center of the given circle: (x - 3)2 + (y + 2)2 = 36 x2 + (y- 5)2 = 9 Example 2 Write the equation of a circle given a radius of 5 and center (3, -1) Write the equation of a circle with a radius of and7a center (-1, 0)