Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Circle Definition Circle : The set of points coplanar points equidistant from a given point. The given point is called the CENTER of the circle. The distance from the center to the circle is called the RADIUS. Center Radius Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Definitions Chord : The segment whose endpoints lie on the circle. Diameter : A chord that contains the center of the circle. Tangent : A line in the plane of the circle that intersects the circle in exactly one point. Tangent Point of Tangency : Chord The point where the tangent line intersects the circle. Diameter Secant : A line that contains a chord. Secant Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Example: In the following figure identify the chords, radii, and diameters. Chords: O D A B F C E Radii: Diameter: Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Definitions Congruent Circles : Circles that have congruent radii. 2 2 Concentric circles : Circles that lie in the same plane and have the same center. Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Polygons Inscribed Polygon: A polygon inside the circle whose vertices lie on the circle. Circumscribed Polygon : A polygon whose sides are tangent to a circle. Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology ARCS Arcs : The part or portion on the circle from some point B to C is called an arc. Named by 2 letters A B C Example: B Semicircle: An arc that is equal to 180°. Named by 3 letters O A Example: C Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Minor Arc & Major Arc Minor Arc : A minor arc is an arc that is less than 180° A minor arc is named using its endpoints with an “arc” above. A Example: Major Arc: A major arc is an arc that is greater than 180°. B B O A major arc is named using its endpoints along with another point on the arc (in order). A Example: C Lesson 8-1: Circle Terminology

Lesson 8-1: Circle Terminology Example: ARCS Identify a minor arc, a major arc, and a semicircle, given that is a diameter. Minor Arc: A C D E F Major Arc: Semicircle: Lesson 8-1: Circle Terminology

Lesson 8-3 Tangents Lesson 8-3: Tangents

THEOREM #1: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Example: Find the value of A B C D 4 3 A B C Lesson 8-3: Tangents

THEOREM #2: If two segments from the same exterior point are tangent to a circle, then they are congruent. B C A Example: Find the value of If AB = 1.8 cm, then AF = 1.8 cm AE = AF + FE AE = 1.8 + 7.0 = 8.8 cm If FE = 7.0 cm, then DE = 7.0 cm CE = CD + DE CE = 2.4 + 7.0 = 9.4 cm 2.4 cm 1.8 7.0 E A C F B D Lesson 8-3: Tangents

Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Theorem #1: In a circle, if two chords are congruent then their corresponding minor arcs are congruent. E A B C D Example: Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Theorem #2: In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. E D A C B Example: If AB = 5 cm, find AE. Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Theorem #3: In a circle, two chords are congruent if and only if they are equidistant from the center. O A B C D F E Example: If AB = 5 cm, find CD. Since AB = CD, CD = 5 cm. Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Try Some Sketches: Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle. Draw a radius so that it forms a right triangle. How could you find the length of the radius? Solution: ∆ODB is a right triangle and 8cm 15cm O A B D x Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Try Some Sketches: Draw a circle with a diameter that is 20 cm long. Draw another chord (parallel to the diameter) that is 14cm long. Find the distance from the smaller chord to the center of the circle. Solution: 10 cm 20cm O A B D C ∆EOB is a right triangle. OB (radius) = 10 cm 14 cm E x 7.1 cm Lesson 8-4: Arcs and Chords

Lesson 8-5: Angle Formulas Angle in Circles Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Central Angle Definition: An angle whose vertex lies on the center of the circle. Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Central Angle Theorem The measure of a center angle is equal to the measure of the intercepted arc. Y Z O 110 Intercepted Arc Center Angle Example: Give is the diameter, find the value of x and y and z in the figure. Lesson 8-5: Angle Formulas

Example: Find the measure of each arc. 4x + 3x + (3x +10) + 2x + (2x-14) = 360° 14x – 4 = 360° 14x = 364° x = 26° 4x = 4(26) = 104° 3x = 3(26) = 78° 3x +10 = 3(26) +10= 88° 2x = 2(26) = 52° 2x – 14 = 2(26) – 14 = 38° Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle). Examples: 3 1 2 4 No! Yes! No! Yes! Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Intercepted Arc Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds: 1. The endpoints of the arc lie on the angle. 2. All points of the arc, except the endpoints, are in the interior of the angle. 3. Each side of the angle contains an endpoint of the arc. Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Inscribed Angle 110 55 Z Intercepted Arc An angle formed by a chord and a tangent can be considered an inscribed angle. Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Examples: Find the value of x and y in the fig. y ° x 50 A B C E F y ° 40 x 50 A B C D E Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas An angle inscribed in a semicircle is a right angle. P 180 90 S R Lesson 8-5: Angle Formulas

Interior Angle Theorem Definition: Angles that are formed by two intersecting chords. 1 A B C D 2 E Interior Angle Theorem: The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs. Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Example: Interior Angle Theorem 91 A C x° y° B D 85 Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. 3 y ° x 2 1 Two secants 2 tangents A secant and a tangent Lesson 8-5: Angle Formulas

Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. Lesson 8-5: Angle Formulas

Example: Exterior Angle Theorem Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Q G F D E C 1 2 3 4 5 6 A 30° 25° 100° Lesson 8-5: Angle Formulas

Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. mDAB + mDCB = 180  mADC + mABC = 180  Lesson 8-5: Angle Formulas

Lesson 8-6: Segment Formulas Segments in Circles Lesson 8-6: Segment Formulas

Intersecting Chords Theorem Interior segments are formed by two intersecting chords. Theorem: If two chords intersect within a circle, then the product of the lengths of the parts of one chord is equal to the product of the lengths of the parts of the second chord. A B C D E a d b c a • b = c • d Lesson 8-6: Segment Formulas

Intersecting Secants/Tangents Exterior segments are formed by two secants, or a secant and a tangent. D B C A D B A C E Secant and a Tangent Two Secants Lesson 8-6: Segment Formulas

Intersecting Secants Theorem If two secant segments are drawn to a circle from an external point, then the products of the lengths of the secant and their exterior parts are equal. a • e = c • f Lesson 8-6: Segment Formulas

Lesson 8-6: Segment Formulas Example: AB  AC = AD  AE D B A C E 4 cm 4  10 = 2  (2+x) 6 cm 2 cm 40 = 4 + 2x x 36 = 2x X = 18 cm Lesson 8-6: Segment Formulas

Secant and Tangent Theorem: The square of the length of the tangent equals the product of the length of the secant and its exterior segment. D B C A a2 = b • d a b c d Lesson 8-6: Segment Formulas

Lesson 8-6: Segment Formulas Example: D B C A x 9 cm 25 cm Lesson 8-6: Segment Formulas