Obj. 37 Proportional Relationships
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The student is able to use ratios to make indirect measurements and use scale drawings to solve problems. Indirect measurement uses formulas, similar figures, and proportions to measure an object. Examples are provided of using ratios and proportions to calculate the height of a flagpole based on shadow lengths, and calculating the height of Reunion Tower based on a mirror reflection. An example of using a scale on a map to calculate the distance between two cities is also provided.
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3.9.4 Indirect Measurement
The student can use ratios and proportions to make indirect measurements from scale drawings or similar figures. An example shows using similar triangles to find the height of a flagpole given the height and shadow of a stick. Another example shows using similar triangles to find the height of Reunion Tower by measuring the angle of reflection in a mirror and distances from the observer's perspective.
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The document discusses similar figures and scale drawings. It provides examples of using proportions to solve problems involving similar figures, scale drawings, and indirect measurements. Key concepts covered include: similar figures having the same shape but not necessarily the same size, with corresponding angles and sides in proportion; using proportions to solve problems involving similar figures and scale drawings; and using similar triangles to indirectly measure quantities like the height of a tree or flagpole using shadow lengths.
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Proportion and ratios (Lucía and Deva- 2ºD)
1. Similar shapes are those that are the same shape but different sizes. Corresponding angles and proportions will be the same between similar shapes. The ratio of similarity is the fixed number by which one shape's sides are multiplied to obtain the other shape.
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This document discusses trigonometry and how it can be used to calculate heights and distances. It defines trigonometric ratios and the angles of elevation and depression. It then provides examples of using trigonometry to calculate the height of a tower given the angle of elevation is 30 degrees and the distance from the observer is 30 meters. It also gives an example of calculating the height of a pole using the angle made by the rope and the ground.
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2. Several exercises involve classifying numbers as rational or irrational, finding upper bounds for absolute and relative errors, and representing numbers on the real number line.
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The document discusses trigonometric applications involving angles of elevation and depression. It defines an angle of elevation as one formed between a horizontal line and a line of sight to a point above the line, and an angle of depression as one formed between a horizontal line and a line of sight to a point below the line. Examples are given to demonstrate using trigonometric functions like tangent to solve for unknown lengths given an angle of elevation or depression and one known side length.
4.12.3 Trig Applications
The document discusses trigonometric applications involving angles of elevation and depression. It defines an angle of elevation as one formed between a horizontal line and a line of sight to a point above the line, and an angle of depression as one formed between a horizontal line and a line of sight to a point below the line. Examples are given to demonstrate using trigonometric functions like tangent to solve for unknown lengths given an angle of elevation or depression and one known side length.
Euclids geometry for class IX by G R Ahmed
This document summarizes key concepts from Euclid's Geometry, including:
1. It discusses Euclid and his introduction of logical reasoning and proof to geometry. He is known for his work on plane figures called "Euclidean Geometry".
2. It defines basic geometric terms like points, lines, planes, and provides Euclid's original definitions and postulates.
3. It covers Euclid's influential axioms and 23 definitions that formed the basis for logic and proof in geometry. The definitions cover concepts like circles, triangles, parallelograms.
Similar to Obj. 37 Proportional Relationships (20)
Measuring tall objects using clinometer, shadow, and mirror method
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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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Obj. 37 Proportional Relationships
- 1. Obj. 37 Proportional Relationships The student is able to (I can): • Use ratios to make indirect measurements • Use scale drawings to solve problems.
- 2. indirect measurement Any method that uses formulas, similar figures, and/or proportions to measure an object. Example: An 8 foot tall stick casts a 6 foot shadow. At the same time, a tall flagpole casts an 18 foot shadow. How tall is the flagpole? x 8 6 18 The triangles are similar by AA~. 8 x 6x = 144 → x = 24 feet = 6 18
- 3. Example Miriam saw a mirror on the ground and noticed that she could see the top of Reunion Tower in the mirror. Her line of sight was 5’ above the ground, and the mirror was 2’ away from her. She measured the distance from that position to the base of Reunion Tower, and it was 224 feet. How high is Reunion Tower? The reflection creates congruent angles, so the triangles are similar by AA~.
- 4. Example x 5’ 2’ 224’ 5 x = 2 224 2x = 1120 x = 560 ft.
- 5. Example According to the map’s scale, 1 in = 100 mi. If it measures 2.75 in. from Ft. Worth to San Antonio, how far away are the two cities? 1 2.75 = 100 x x = 275 mi.