Obj. 55 Lines and Circles
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The student can identify and use properties of chords, tangents, and secants of circles. This includes using properties of tangents to solve problems, recognizing that congruent circles have congruent radii, defining concentric and tangent circles, and understanding theorems relating tangents, radii, and segments tangent from the same external point to a circle. The student can apply these concepts to find values in example problems involving tangent segments.
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The document defines key terms related to circles and tangents, including: a circle as points equidistant from a center point; a radius and diameter; a chord as a segment with endpoints on the circle; a secant as a line intersecting a circle at two points; and a tangent as a line intersecting a circle at exactly one point. It states that a tangent line is perpendicular to the radius drawn to the point of tangency, and that for two segments from the same exterior point to both be tangent to a circle, they must be congruent.
RELATED TERMS TO CIRCLE(Radius, diameter, etc)
This document contains definitions and examples related to circles:
1. It defines a circle as the set of all points equidistant from a center point and defines related terms like radius, diameter, chord, secant, and tangent.
2. Examples of circles in real life are given like donuts, rubber bands, and rings.
3. Diagrams are included labeling the parts of a circle like the radius, diameter, chord, secant, and tangent. Definitions are provided for each term.
Circles
Here are the steps to solve this problem:
1) Since ED is tangent at C, ∠ECD = 90° (property of tangents)
2) ∆OCD ~ ∆OED (AA similarity)
3) Corresponding parts are proportional:
OC/OD = OC/OE
Cross multiply: OC×OE = OD×OC
Divide both sides by OC: OE = OD
4) Given DE = 20, OE = OD = 20/2 = 10
5) ∠ACD = ∠AED (inscribed angles on the same arc)
6) ∆ACD ~ ∆AED (AA similarity)
7) Corresponding parts are proportional
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Circles day 3 segments and other angles and more2013
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* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
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* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
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3.1 Functions
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* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
2.5 Transformations of Functions
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* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
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2.2 More on Functions and Their Graphs
This document discusses functions and their graphs. It defines increasing, decreasing and constant functions based on how the function values change as the input increases. Relative maxima and minima are points where a function changes from increasing to decreasing. Symmetry of functions is classified by the y-axis, x-axis and origin. Even functions are symmetric about the y-axis, odd functions are symmetric about the origin. Piecewise functions have different definitions over different intervals.
1.6 Other Types of Equations
This document provides examples and steps for solving various types of equations beyond linear equations, including:
1) Polynomial equations solved by factoring
2) Equations with radicals where radicals are eliminated by raising both sides to a power
3) Equations with rational exponents where both sides are raised to the reciprocal power
4) Equations quadratic in form where an algebraic substitution is made to transform into a quadratic equation
5) Absolute value equations where both positive and negative solutions must be considered.
1.5 Quadratic Equations (Review)
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2.1 Basics of Functions and Their Graphs
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- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
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13.2 Independent & Dependent Events
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* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
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9.4 Series and Their Notations
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* Solve annuity problems.
9.3 Geometric Sequences
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* List the terms of a geometric sequence.
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Obj. 55 Lines and Circles
- 1. Obj. 55 Lines and Circles The student is able to (I can): • Identify chords, tangents, and secants • Use properties of tangents to solve problems
- 2. secant tangent point of tangency A line that intersects a circle at two points A line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect. • A B m C chord secant tangent point of tangency
- 3. congruent circles Two circles are congruent if and only if their radii are congruent. A B C D A ≅ C if AB CD≅ AB CD≅ if A ≅ C
- 4. concentric circles tangent circles Coplanar circles with the same center Two coplanar circles that intersect at exactly one point. • internally tangent externally tangent
- 5. common tangent A line that is tangent to two circles. common external tangent common internal tangent
- 6. Thm 11-1-1 Thm 11-1-2 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 at a point on the circle, then it is tangent to the circle. BELine t ⊥ Line t tangent to B •B E t
- 7. Thm 11-1-3 If two segments are tangent to a circle from the same external point, then the two segments are congruent. • S A N D SD ND≅
- 8. Examples The segments in each figure are tangent. Find the value of each variable. 1. 2. • 2a + 4 5a — 32 • 6y2 18y 5a — 32 = 2a + 4 3a = 36 a = 12 6y2 = 18y y y 6y = 18 y = 3