Chapter 1 3

Page 1: Tangents to Circles

Objective

  • To find missing information related to parts of a circle.

  • Do Now (DOL): I will correctly solve 4 problems.

Definitions

  • Circle: A set of all points in a plane that are equidistant from a point called the center.

  • Chord: A segment whose endpoints lie on the circle.

  • Diameter: A special chord that passes through the center.

  • Radius: Half of the diameter; the distance from the center to a point on the circle.

  • Equal Circles: Two circles are congruent if they have the same radius.

  • Secant: A line that intersects a circle at two points.

  • Tangent: A line that intersects a circle at exactly one point (point of tangency).

Identifying Circle Parts

List of segments to classify:

  1. AG

  2. CB

  3. ED

  4. AB

  5. AG

Circle Intersections

  • Intersecting Circles: Two points

  • Tangent Circles: One point of intersection (tangent)

  • Concentric Circles: No intersections (same center)

Classification of Segments

  • A segment whose endpoints are on the circle is a chord.

  • A chord that goes through the center is a diameter.

Page 2: Theorems

Key Theorems

  • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

  • If a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is tangent to the circle.

  • If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example Problems

  1. Check if EF is tangent to circle D.

  2. Find AC if BC is tangent to circle A.

  3. AB and AD are tangents to circle C; find x.

Practice Problems:

  • Use formulas and given values to solve:

    • 9; 23a^2 + b = 612

    • 112 + 602

    • Yes check; 3721.

    • Solve for x in triangles:

      • x = 26; x + 2 = 1.

      • xz = 9; x = 3.

Page 3: Practice Problems

Problems

  1. AB is tangent to circle C; find x.

  2. Check if CE is tangent to circle D; find the radius.

  3. BC is tangent to circle A.

Algebraic Solutions

  • Solve quadratic equations like:

    • xz - 4 = 2 (to x).

    • x^2 = 25; simplify for values.

Page 4: Finding Variables

Key Insights

  • Lines that appear tangent are treated as tangent for calculations.

  • Solve for missing variables using tangent relationships.

  • Example degenerate cases of polynomial equations.

  • Conclude and validate through solving quadratics.

Example Problems

  1. Given equations like x^2 - 5x = 3; find x.

  2. y^2 + 36.

Page 5: Extra Practice

Circle and Angle Problems

  • Provided dimensions:

    • AB = 12, DE = 12, CE = 7; find CG.

  • Radius considerations simplify relationships:

  1. If m ∠ 64°, find mABD.

  2. If mAB = 108°, derive m ∠ 3.

  3. If m BDA = 290°, derive m ∠ 1.

  4. If m = 137°, find mACD.

Page 6: Additional Angles

Angle Measures

  • Examples:

  1. If m∠ = 60° and BD = 16, find AE.

  2. If m∠1 = 30° and KJ = 9, find BD.

  3. If KE = 12 and BD = 10, find KM.

  • Answers for practice include finding intersections and their values based on existing angles.

Page 7: More Practice

Example Problems

  1. Find mAD.

  2. Given AD = 40, CD = 25, find CG.

  3. Calculate the length of a chord given distance from the center and radius.

Arc Classifications

  • Determine whether arcs are major, minor, or semicircle based on angle measures in the circle.

  • Evaluate and solve for arc lengths based on circle properties.

Page 8: Continued Practice

  • Solve for mBC and mDC. Deriving values through tangent relationships and properties of chords.

  • Example lengths for radius and distance interactions using chord properties and calculations:

  1. Length of radius, chord, and associated measures.

Page 9: Extra Practice

Circle Measurements

  • Given that E is the circle center, find angle measures through derived relationships and intersection angles.

  • Check diameter use in angle calculations and confirm with known values.

Page 10: More Angle Relationships

Angle Measures

  • Finding the measures of angles using tangents and segments through chord intersection points.

  • Evaluating relationships and solving similar triangles.

Page 11: Inscribed Angles

Definitions and Objectives

  • Inscribed angle: Vertex on the circle, intersecting chords.

  • Theorem: If an angle inscribed in a circle, the angle measure is half the arc measure.

Examples

a) Find mSTQ. b) Find mZWX. c) Calculate m∠NMP. d) m∠B - A C B.

Page 12: Inscribed Polygon

Finding Variables

  • Theorems related to inscribed angles and supplementary relationships.

  • Practice intercept angles and solve simultaneous equations to find measures.

Page 13: Further Practice

Additional Measures

  • Find angle measures through algebraic representation and solving for x, focusing on inscribed angles.

Page 14: Extra Practice

  • Solve equations and find generalized values that relate to angle measures in circle segments.

  • Example solving systems leading to variable intersections.

Page 15: Final Angle Measures

Additional Angle Facts

  • Given angles in a circle and deriving arcs and external angles through relationships.

Page 16: Other Angle Relationships

Objective

  • Determine angle or arc measures from segments.

Key Theorems

  1. Tangent and Chord intersecting: Measure of the angle is ½ the arc.

  2. Angles formed by intersections are averages of arc measures.

Page 17: Exterior Circles

  1. Two tangent angles: The measure of angles formed from external graphical representations.

  2. Solve for missing angles using properties of secants and tangents in geometry equations.

Page 18: Practice Interactions

Solving Variables

  • Address external angles based on secant and tangent ratio calculations.

  • Solve geometrically for visible tangents, equivalently representing x and solving quadratic systems based on established equations.

Page 19: Further Interactions

  • Create linear equations based on established circle theory, solving for variables through defined tangents and secants.

Page 20: Angle Solutions

Analyzing Relationships

  • Utilize established equations to solve for variables in the angles formed from chords, secants, and tangency conditions.

  • Address the relationships through systematic evaluations.

Page 21: Geometry Review

Practice Problems

  1. Find the measure of angles and create equations for variables based on descriptions.

  2. Use diagrams to evaluate order and conditions for solving.

Page 22: Circle Relationships

Given Measurements

  • Evaluate angle relationships and find all variables based on established rules from previous exercises.

  • Solve for specific arc measures and angle conditions.