Rational Expressions Day 6
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This document discusses methods for solving fractional equations including cross multiplication, simplifying expressions, and multiplying by the least common denominator. It provides examples of each method and includes a practice section with two problems to solve involving determining undefined values and factoring completely.
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Notes Day 6: Bernoulli Trials
This document defines Bernoulli trials and the binomial distribution. A Bernoulli trial is a random experiment with two possible outcomes: success or failure. A sequence of Bernoulli trials involves performing the experiment multiple independent times, keeping the probability of success the same. The binomial distribution calculates the probability of getting a certain number of successes in n Bernoulli trials. It satisfies properties where the experiment is conducted n times independently with a constant success probability p and failure probability of 1-p.
Notes Day 6: Prove that sides are in proportion
This document appears to be an attachment of classwork on corresponding sides of similar triangles. The attachment is a PDF file that likely contains exercises, examples, or instructions related to teaching students about corresponding sides of similar triangles in geometry. In fewer than 3 sentences, the document provides homework or class material on similar triangles and their corresponding sides.
Notes Day 4&5: Prove that Triangles are Similar
The document discusses three theorems for determining similarity of triangles:
1) The Angle-Angle Similarity (AA) Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2) The Side-Angle-Side Similarity (SAS) Theorem states that if an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.
3) The Side-Side-Side Similarity (SSS) Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
Day 13-3: Similar Polygons
Similar figures have the same shape but not necessarily the same size. Two polygons are similar if their corresponding angles are congruent and their corresponding sides are proportional. The symbol for similarity is ~. This document provides examples of determining if shapes are similar by checking if corresponding angles are congruent and sides are proportional.
Day 14-4: Probability with Permutations & Combinations
This document provides examples of using permutations and combinations to calculate probabilities of events occurring. It discusses:
- Calculating the probability of selecting two white and two red balls from an urn containing six white and five red balls by finding the number of ways to choose two white and two red balls, and dividing by the total number of possible outcomes.
- Calculating probabilities for other events involving selecting balls from urns with different ball combinations.
- Calculating probabilities for events involving selecting students for a team from groups with different numbers of boys and girls.
- Calculating the probability of choosing a specific letter from a word by counting the number of ways to choose that letter and dividing by the total letters.
Probability Day 3 - Permutations and Combinations
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Probability day 1
The homework assignment includes problems from two pages in the textbook, with problems 1, 2, 10-14, and 24-27 from pages 42-43, and problems 1-17 odd and 36-40 from pages 534-536.
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This document defines Bernoulli trials and the binomial distribution. A Bernoulli trial is a random experiment with two possible outcomes: success or failure. A sequence of Bernoulli trials involves performing the experiment multiple independent times, keeping the probability of success the same. The binomial distribution calculates the probability of getting a certain number of successes in n Bernoulli trials. It satisfies properties where the experiment is conducted n times independently with a constant success probability p and failure probability of 1-p.
Notes Day 6: Prove that sides are in proportion
This document appears to be an attachment of classwork on corresponding sides of similar triangles. The attachment is a PDF file that likely contains exercises, examples, or instructions related to teaching students about corresponding sides of similar triangles in geometry. In fewer than 3 sentences, the document provides homework or class material on similar triangles and their corresponding sides.
Notes Day 4&5: Prove that Triangles are Similar
The document discusses three theorems for determining similarity of triangles:
1) The Angle-Angle Similarity (AA) Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2) The Side-Angle-Side Similarity (SAS) Theorem states that if an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.
3) The Side-Side-Side Similarity (SSS) Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
Day 13-3: Similar Polygons
Similar figures have the same shape but not necessarily the same size. Two polygons are similar if their corresponding angles are congruent and their corresponding sides are proportional. The symbol for similarity is ~. This document provides examples of determining if shapes are similar by checking if corresponding angles are congruent and sides are proportional.
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This document provides examples of using permutations and combinations to calculate probabilities of events occurring. It discusses:
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- Calculating probabilities for other events involving selecting balls from urns with different ball combinations.
- Calculating probabilities for events involving selecting students for a team from groups with different numbers of boys and girls.
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Probability Day 3 - Permutations and Combinations
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Lsn 12 6
This document contains a graph with labeled x and y axes ranging from -6 to 6. Within this range are curved lines forming a circle centered at the point (4, 1).
Day 2 And 3 Midpoint And Slope
The document provides formulas and examples for calculating slope and midpoints of lines and determining properties of triangles. It includes the slope formula, definitions for lines with the same or opposite reciprocal slopes, the midpoint formula, an example of finding the other endpoint given one endpoint and the midpoint, and an example of proving a triangle is isosceles and right using its vertex coordinates.
Lsn 12-1: The Distance Formula
To find the distance between points G(-5, -8) and H(3,7) use the distance formula. Also use the distance formula to find the distance between points L(15, 37) and M(42, 73). The distance formula is used to calculate the distance between the points (-19, -16) and (-3, 14).
Lsn 13-1: Function Translations
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Lsn 11 12 Kites
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Lsn 11-12: Kites
This document contains numerical values for variables a, b, c, and d in two sections about arithmetic and midsegments. In the arithmetic section, the values are 42, 63, 117, and 14 for the first line, and 30, 28.5, 87, and 12 for the second line. In the algebra section, x is given as 6 for the first two lines, but no values are provided for the third line.
Lsn 11-10: Applications & Regents Problems
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Lsn 11-6: Law of Cosines
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Lsn 11-7: Law of Cosines to find an Angle
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Lsn 11-5: Law of Sines
Here are the steps to solve these Law of Sines problems:
1. Given: a = 13, b = 20, A = 75°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(75°)/13 = sin(B)/20
sin(75°)*20/13 = sin(B)
B = sin-1(0.8) = 67°
2. Given: a = 25, B = 38°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(38°)/25 = sin(m°)/20
sin(38°)*20/25 = sin
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To prove that a quadrilateral is a rectangle or rhombus, you must first prove that it is a parallelogram by showing that both pairs of opposite sides are parallel and congruent. Then, you need only one additional condition - either that the diagonals are congruent (rectangle) or that one angle is a right angle (rectangle), or that the diagonals are the same length (rhombus) or that one pair of consecutive sides are congruent (rhombus). The document provides an example proof of showing a quadrilateral is a rectangle.
Lsn 11-7 Proofs Given Rhombii
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This document appears to be notes from math lessons covering solving second degree trigonometric equations and examples. It includes solving an equation for theta, and working through an example problem.
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This document contains a series of numbers and letters arranged in a graphical format with some numbers assigned to the variables x. It shows the values 7 assigned to x in several locations and the value -3 assigned to x in one location. The document also includes four line items labeled 1 through 4 at the bottom.
Lsn 11-1
1) Inverse trig functions like arcsin, arccos, and arctan are used to find missing angles when given sides of a right triangle.
2) On a TI calculator, these inverse trig functions can be found using synonyms like sin^-1, cos^-1, and tan^-1.
3) Examples show how to use inverse trig functions to find angles in one or two steps, both with and without a calculator.
Regents Trig Graphing Problems
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Lsn 11-3 Proving a Quadrilateral is a Parallelogram
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Lsn 12 6
This document contains a graph with labeled x and y axes ranging from -6 to 6. Within this range are curved lines forming a circle centered at the point (4, 1).
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The document provides formulas and examples for calculating slope and midpoints of lines and determining properties of triangles. It includes the slope formula, definitions for lines with the same or opposite reciprocal slopes, the midpoint formula, an example of finding the other endpoint given one endpoint and the midpoint, and an example of proving a triangle is isosceles and right using its vertex coordinates.
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To find the distance between points G(-5, -8) and H(3,7) use the distance formula. Also use the distance formula to find the distance between points L(15, 37) and M(42, 73). The distance formula is used to calculate the distance between the points (-19, -16) and (-3, 14).
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The passage discusses the importance of summarization in efficiently conveying key information from lengthy documents. It notes that effective summaries distill the most critical details and events into a brief yet informative recap of the overall topic and narrative. The ability to produce high-quality summaries is a useful skill for professionals across many fields in order to save time and focus on the most pertinent details.
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This document contains numerical values for variables a, b, c, and d in two sections about arithmetic and midsegments. In the arithmetic section, the values are 42, 63, 117, and 14 for the first line, and 30, 28.5, 87, and 12 for the second line. In the algebra section, x is given as 6 for the first two lines, but no values are provided for the third line.
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This document contains 3 trigonometry word problems: 1) Find the length of a segment labeled x in a diagram with a 12 foot side. 2) Given a 46 foot sign on a building, find the height of the building using angles of elevation. 3) Given angles of elevation to a lighthouse from a ship and its distance traveled, find how close the ship is to dangerous rocks near the lighthouse.
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The document provides examples of using the Law of Sines and Law of Cosines to solve for missing sides and angles of triangles. It includes 5 multi-step triangle problems worked out using the two trigonometric laws, with the key points being that Law of Sines is used when an angle is given alone, Law of Cosines is used when a side is given alone or one triangle has a right angle, and letter assignments in the Law of Cosines formula can be changed.
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The document provides 3 examples of using the Law of Cosines to find the measure of an angle given the lengths of 3 sides of a triangle. The first example finds the measure of an acute angle C in a triangle with sides of lengths 5, 12, and 9. The second finds the right angle C in a triangle with sides of lengths 5, 12, and 13. The third finds the measure of an obtuse angle C in a triangle with sides of lengths 5, 12, and 15.
Lsn 11-5: Law of Sines
Here are the steps to solve these Law of Sines problems:
1. Given: a = 13, b = 20, A = 75°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(75°)/13 = sin(B)/20
sin(75°)*20/13 = sin(B)
B = sin-1(0.8) = 67°
2. Given: a = 25, B = 38°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(38°)/25 = sin(m°)/20
sin(38°)*20/25 = sin
Lsn 11-8: Proving a Rhombus/Rectangle
To prove that a quadrilateral is a rectangle or rhombus, you must first prove that it is a parallelogram by showing that both pairs of opposite sides are parallel and congruent. Then, you need only one additional condition - either that the diagonals are congruent (rectangle) or that one angle is a right angle (rectangle), or that the diagonals are the same length (rhombus) or that one pair of consecutive sides are congruent (rhombus). The document provides an example proof of showing a quadrilateral is a rectangle.
Lsn 11-7 Proofs Given Rhombii
O documento apresenta várias equações matemáticas com símbolos de igualdade e desigualdade relacionando números e medidas de área. As equações definem relações entre diferentes variáveis numéricas e unidades de medida.
Lsn 11-3 Solving Second Degree Trig Equations
This document appears to be notes from math lessons covering solving second degree trigonometric equations and examples. It includes solving an equation for theta, and working through an example problem.
Lsn 11-4 Area of a Triangle Formula
The document contains instructions and examples for calculating the area of triangles using various formulas. It asks the reader to find the area of triangles given dimensions and angle measures. It also provides the formula for finding the area of a parallelogram given the base, height, and angle between the base and height.
Lsn 11-2: Solving Trig Equations
This document provides examples for solving trigonometric equations. It discusses that each trig function is positive a certain number of times and negative a certain number of times from 0 to 360 degrees. It explains that calculators only provide one answer but there may be multiple answers, and using a reference angle from the first quadrant allows finding other answers. It then provides 5 examples of solving trig equations, finding where trig functions are equal to given values, and checking solutions on a calculator.
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This document contains a series of numbers and letters arranged in a graphical format with some numbers assigned to the variables x. It shows the values 7 assigned to x in several locations and the value -3 assigned to x in one location. The document also includes four line items labeled 1 through 4 at the bottom.
Lsn 11-1
1) Inverse trig functions like arcsin, arccos, and arctan are used to find missing angles when given sides of a right triangle.
2) On a TI calculator, these inverse trig functions can be found using synonyms like sin^-1, cos^-1, and tan^-1.
3) Examples show how to use inverse trig functions to find angles in one or two steps, both with and without a calculator.
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This document contains calculations to find the average (mean), the period, and a value shifted to the right by 47 days for data that is periodic with a period of 365 days. It calculates the average as (maximum value + minimum value)/2, the period directly as 365 days, and a value shifted 47 days to the right as 2.52 within the periodic range.
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There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
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Rational Expressions Day 6
- 1. January 19, 2010 Lsn 86: Solving Fractional Equations 1. Cross Multiplication: ex.: (restriction: )