6.4 Triangle Midsegments
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The document discusses triangle midsegments. A midsegment is a segment that joins the midpoints of two sides of a triangle. The Triangle Midsegment Theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Examples are provided to demonstrate using the theorem to find the measures of lengths and angles associated with midsegments in triangles.
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4.10.3 Triangle Midsegment Theorem
The midsegment theorem states that a midsegment of a triangle, which is a segment that joins the midpoints of two sides, is parallel to the opposite side and its length is half the length of that side. The document provides examples of using the midsegment theorem to find lengths of midsegments and measure of angles related to midsegments. It defines a midsegment and illustrates it in diagrams of triangles.
2.5.3 Triangle Midsegments
The document discusses midsegments of triangles. A midsegment is a segment that joins the midpoints of two sides of a triangle. The triangle midsegment theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Examples are provided to demonstrate finding the measures of midsegments and related angles using properties of midsegments.
2.5.7 Triangle Midsegments
The document discusses triangle midsegments. A midsegment is a segment that joins the midpoints of two sides of a triangle. The Triangle Midsegment Theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Examples are provided to demonstrate using the theorem to find the measures of lengths and angles associated with midsegments in triangles.
2.5.7 Medians and Altitudes
* Identify altitudes and medians of triangles
* Identify the orthocenter and centroid of a triangle
* Use triangle segments to solve problems
2.5.6 Medians and Altitudes
Identify altitudes and medians of triangles
Identify the orthocenter and centroid of a triangle
Use triangle segments to solve problems.
Obj. 21 Medians, Altitudes, and Midsegments
Identify altitudes and medians of triangles
Identify the orthocenter and centroid of a triangle
Use triangle segments to solve problems
Identify a midsegment of a triangle and use it to solve problems.
5.1 midsegment
This document provides instructions for a lesson on midsegments in triangles. It defines what a midsegment is, notes that triangles have three midsegments, and outlines a lab for students to complete in groups to learn about the midsegment theorem. The theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. The document includes example problems for students to practice applying the midsegment theorem.
Obj. 23 Triangle Theorems
Identify a midsegment of a triangle and use it to solve problems.
Analyze the relationship between the angles of a triangle and the lengths of the sides
Determine allowable lengths for sides of triangles
Recommended
4.10.3 Triangle Midsegment Theorem
The midsegment theorem states that a midsegment of a triangle, which is a segment that joins the midpoints of two sides, is parallel to the opposite side and its length is half the length of that side. The document provides examples of using the midsegment theorem to find lengths of midsegments and measure of angles related to midsegments. It defines a midsegment and illustrates it in diagrams of triangles.
2.5.3 Triangle Midsegments
The document discusses midsegments of triangles. A midsegment is a segment that joins the midpoints of two sides of a triangle. The triangle midsegment theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Examples are provided to demonstrate finding the measures of midsegments and related angles using properties of midsegments.
2.5.7 Triangle Midsegments
The document discusses triangle midsegments. A midsegment is a segment that joins the midpoints of two sides of a triangle. The Triangle Midsegment Theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Examples are provided to demonstrate using the theorem to find the measures of lengths and angles associated with midsegments in triangles.
2.5.7 Medians and Altitudes
* Identify altitudes and medians of triangles
* Identify the orthocenter and centroid of a triangle
* Use triangle segments to solve problems
2.5.6 Medians and Altitudes
Identify altitudes and medians of triangles
Identify the orthocenter and centroid of a triangle
Use triangle segments to solve problems.
Obj. 21 Medians, Altitudes, and Midsegments
Identify altitudes and medians of triangles
Identify the orthocenter and centroid of a triangle
Use triangle segments to solve problems
Identify a midsegment of a triangle and use it to solve problems.
5.1 midsegment
This document provides instructions for a lesson on midsegments in triangles. It defines what a midsegment is, notes that triangles have three midsegments, and outlines a lab for students to complete in groups to learn about the midsegment theorem. The theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. The document includes example problems for students to practice applying the midsegment theorem.
Obj. 23 Triangle Theorems
Identify a midsegment of a triangle and use it to solve problems.
Analyze the relationship between the angles of a triangle and the lengths of the sides
Determine allowable lengths for sides of triangles
6.7 Exponential and Logarithmic Models
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
4.5 Special Segments in Triangles
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
1.4 Conditional Statements
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
1.3 Distance and Midpoint Formulas
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
1.5 Quadratic Equations.pdf
This document provides instruction on factoring polynomials and quadratic equations. It begins by reviewing factoring techniques like finding the greatest common factor and factoring trinomials and binomials. Examples are provided to demonstrate the factoring methods. The document then discusses solving quadratic equations by factoring, putting the equation in standard form, and setting each factor equal to zero. An example problem demonstrates solving a quadratic equation through factoring. The document concludes by assigning homework and an optional reading for the next class.
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.1 Functions
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
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* Recognize graphs of common functions.
* 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.
* Combine transformations.
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)
This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- 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
9.6 Binomial Theorem
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.
13.3 Venn Diagrams & Two-Way Tables
The document discusses using Venn diagrams and two-way tables to organize data and calculate probabilities. It provides examples of completing Venn diagrams and two-way tables based on survey data about students' activities. It then uses the tables and diagrams to calculate probabilities of different outcomes. The examples illustrate how to set up and use these visual representations of categorical data.
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* Identify events as independent or dependent
* Calculate the probabilities of independent and dependent events
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* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
13.1 Geometric Probability
* Calculate geometric probabilities
* Use geometric probability to predict results in real world situations.
9.4 Series and Their Notations
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
9.3 Geometric Sequences
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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6.7 Exponential and Logarithmic Models
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
4.5 Special Segments in Triangles
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
1.4 Conditional Statements
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
1.3 Distance and Midpoint Formulas
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
1.5 Quadratic Equations.pdf
This document provides instruction on factoring polynomials and quadratic equations. It begins by reviewing factoring techniques like finding the greatest common factor and factoring trinomials and binomials. Examples are provided to demonstrate the factoring methods. The document then discusses solving quadratic equations by factoring, putting the equation in standard form, and setting each factor equal to zero. An example problem demonstrates solving a quadratic equation through factoring. The document concludes by assigning homework and an optional reading for the next class.
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.1 Functions
* Introduce functions and function notation
* 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
* Recognize graphs of common functions.
* 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.
* Combine transformations.
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)
This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- 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
9.6 Binomial Theorem
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.
13.3 Venn Diagrams & Two-Way Tables
The document discusses using Venn diagrams and two-way tables to organize data and calculate probabilities. It provides examples of completing Venn diagrams and two-way tables based on survey data about students' activities. It then uses the tables and diagrams to calculate probabilities of different outcomes. The examples illustrate how to set up and use these visual representations of categorical data.
13.2 Independent & Dependent Events
* Identify events as independent or dependent
* Calculate the probabilities of independent and dependent events
9.5 Counting Principles
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
13.1 Geometric Probability
* Calculate geometric probabilities
* Use geometric probability to predict results in real world situations.
9.4 Series and Their Notations
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
9.3 Geometric Sequences
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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6.4 Triangle Midsegments
- 1. Triangle Midsegments The student is able to (I can): • Identify a midsegment of a triangle and use it to solve problems.
- 2. midsegmentmidsegmentmidsegmentmidsegment –––– a segment that joins the midpoints of two sides of a triangle. H O T I C E Points I, C, and E are midpoints of ΔHOT. , ,andIC CE EI are midsegments.
- 3. Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. H O T I C E 1 , 2 IC HT HT=IC
- 4. Examples Find each measure. 1. FE 2. m∠UFE 3. UE F L U T E S 14141414 62626262°°°° andLT TS are midsegments. 9
- 5. Examples Find each measure. 1. FE FE = 2(LT) = 2(14) = 28 2. m∠UFE m∠UFE = m∠TSE = 62° 3. UE UE = 2(9) = 18 F L U T E S 14141414 62626262°°°° andLT TS are midsegments. 9