Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Euler angles wikipedia , lookup
Approximations of π wikipedia , lookup
Rational trigonometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Problem of Apollonius wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Area of a circle wikipedia , lookup
Circles Chapter 9 Tangent Lines (9-1) • A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. • The point where a circle and a tangent intersect is the point of tangency. B A P Tangent Lines (9-2) • Theorem: If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. B A P Tangent Lines (9-2) • Converse: If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. B A A P 8 4 B 7 D Tangent Lines (9-2) • Corollary: The two segments tangent to a circle from a point outside the circle are congruent. • AB = BC A Q B C Tangent Lines (9-1) B • “Inscribed in the circle ” C A • “Circumscribed about the circle” S T U R V Tangent Lines (9-1) • Circle G is inscribed in quadrilateral CDEF. Find the perimeter of CDEF. D 8 ft E 6 ft 11 ft G 7 ft C F Arcs and Central Angles 9-3 • Central Angle (of a circle)- angle with its vertex at the center of the circle • Arc- unbroken part of a circle • Minor Arc (less than 180 degrees) • Name them using the endpoints • Major Arc (more than 180 degrees) • Name them using three points • Semicircles- two arcs formed by the endpoints of a diameter Arcs and Central Angles 9-3 • Measure of a minor arc= measure of its central angle • Measure of a major arc= 360 degrees – measure of its minor arc • Adjacent arcs- arcs with exactly one point in common (crust of adjacent pizza slices) Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs • Similar to the Angle Addition Postulate Congruent Arcs • Arcs in the same circle or congruent circles • Have equal measures • Arcs in two circles of different sizes cannot be congruent, even if they have the same measure (to be congruent, they must be the same shape and size) Theorem 9-3 • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent • STOP Chords and Arcs (9-4) • A chord is a segment whose endpoints are on a circle. • Each chord cuts off a minor arc and a major arc B C A Chords and Arcs (9-4) • Theorem: Within a circle or congruent circles 1. Congruent arcs have congruent chords. 2. Congruent chords have congruent arcs. Chords and Arcs (9-4) • Within a circle or in congruent circles… B C A D Theorem 9-5 • A diameter that is perpendicular to a chord bisects the chord and its arc. Converse… • In a circle, a diameter that bisects a chord (that is not the diameter) is perpendicular to the chord. • Example 86 degrees Chords and Arcs (9-3) • Theorem: Within a circle or congruent circles 1. Chords equidistant from the center are congruent. 2. Congruent chords are equidistant from the center. Chords and Arcs (9-4) • Find x. Chords and Arcs (9-4) • Find HL and QJ. H 11 26 Q J L • HL= 22, QJ = 4 √3 Chords and Arcs (9-4) • In a circle, the perpendicular bisector of a chord contains the center of the circle. • STOP Inscribed Angles (9-5) • Inscribed angle – vertex on the circle, sides of angle are chords of circle • Intercepted arc – arc formed when the sides of the inscribed angle cross the circle A C B Inscribed Angles (9-5) • Theorem: The measure of an inscribed angle is half the measure of its intercepted arc. A C B Inscribed Angles (9-5) D 80 • Find x and y. 70 A y • x= ½ *(80+70) • x= 75° • m arc BC= 360- (80+70+90) = 120° • y= ½ * (70+120)= 95° C 90 x B Inscribed Angles (9-5) • Corollary- Two inscribed angles that intercept the same arc are congruent. Inscribed Angles (9-5) Corollary- An angle inscribed in a semicircle is a right angle. • GeoGebra example Inscribed Angles (9-5) Corollary- The opposite angles of a quadrilateral inscribed in a circle are supplementary. Inscribed Angles (9-5) • Find the value of a and b. • a= 90° • 2 *32° = 64° • b= 180- 64= 116° 32 E b a • 9-5 handout • Problems 1-9 all Inscribed Angles (9-5) • The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. I F G H • 9-5 handout • Problems 10-21 all Angle Measure and Segment Lengths (9-5) • A secant is a line that intersects a circle at two points. A B Angle Measure and Segment Lengths (9-6) • The measure of an angle formed by two lines that intersect 1. inside a circle is half the sum of the measures of the intercepted arcs. 2. outside a circle is half the difference of the measure of the intercepted arcs. The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measure of the intercepted arcs • Find the measure of <1 • m<1= ½ (45 + 75) • = 60 The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measure of the intercepted arcs. • m <B = ½ (m AFD - m AC) • 65 = ½ (m AFD – 70) • 200 = m AFD Angle Measure and Segment Lengths (9-6) • Find the value of x. • x = ½ (268 – 92) • x = 88 268 92 x Angle Measure and Segment Lengths (9-6) • • • • Find the value of x. 94 = ½ (x + 122) 188 = x + 122 x = 66 x 94 112 Angle Measure and Segment Lengths (9-6) Where the angle vertex is Center of circle Angle measure m(arc) On circle ½ m(arc) Inside circle ½ sum of m(arcs) Outside circle ½ difference of m(arcs) Angle Measure and Segment Lengths (9-7) ab=cd a c x b w d y z (w + x)w = (y + z)y t y z (y + z)y = t2 Angle Measure and Segment Lengths (9-7) • Find the value of x. 7 5 3 x Angle Measure and Segment Lengths (9-7) • Find the value of y. 15 8 y Angle Measure and Segment Lengths (9-7) 90 O x a 8 30 b 34 8 12 Angle Measure and Segment Lengths (9-7) 98 8 7 x M a 21