PROPORTIONALITY THEOREM.pptx
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This document discusses proportionality theorems and special right triangle theorems. It then provides 3 learning tasks involving solving word problems about special right triangles, asking for lengths of sides and heights in simplest radical form. The first problem involves a 45-45-90 right triangle with a base of 6 ft, asking for the height and hypotenuse. The second problem involves finding the height of an equilateral triangle with a side length of 34 inches.
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3) When dividing powers with the same base, subtract the exponents.
4) When raising a power to another power, multiply the exponents.
5) When taking a power of a product, apply the exponent to each factor.
6) When taking a power of a quotient, apply the exponent to the numerator and denominator.
7) A negative exponent changes the base to its reciprocal with a positive exponent.
8) Any base raised to the zero power equals one.
secant.ppt
This document defines various circle terms like tangent, secant, concentric circles, and common tangents. It then discusses theorems for finding angle measures based on whether the vertex is on the circle, inside the circle, or outside the circle. For a vertex on the circle, the angle measure is half the intercepted arc. For inside, it's half the sum of intercepted arcs. For outside, it's half the difference of intercepted arcs. Examples are provided to demonstrate applying these theorems to find unknown angle measures.
Exponents Intro with Practice.ppt
The document discusses exponents and laws regarding operations with exponents. It defines exponents as indicating the number of times a base is multiplied by itself. The laws covered include: keeping the base and adding exponents when multiplying powers; keeping the base and subtracting exponents when dividing powers; multiplying exponents when raising a power to an exponent; applying the exponent to all factors when taking a power of a product or quotient; changing the base to its reciprocal and making the exponent positive when the exponent is negative; and any base with a zero exponent equals one. Examples are provided to demonstrate how to apply each law.
Exponents Intro with Practice.ppt
This document defines exponents and how they are used to represent repeated multiplication of a base number. Exponents tell how many times the base number is multiplied by itself. The exponent stands for the number of times the base is multiplied. Examples are provided of how to read exponents aloud and write them in standard form by multiplying out the base the number of times indicated by the exponent.
altitude to the right triangle.pptx
The document discusses the geometric mean of numbers and the properties of right triangles when an altitude is drawn to the hypotenuse. It states that the altitude is the geometric mean between the hypotenuse segments and that each leg is the geometric mean between the hypotenuse and adjacent hypotenuse segment. It also provides an example problem about finding distances between a person and objects in a room.
circle theorem 2.ppt
This document outlines 8 circle theorems: 1) The angle at the center is twice the angle at the circumference. 2) The angle in a semi-circle is 90 degrees. 3) Angles in the same segment are equal. 4) Opposite angles in a cyclic quadrilateral add up to 180 degrees. 5) The lengths of the two tangents from a point to a circle are equal. 6) The angle between a tangent and a radius in a circle is 90 degrees. 7) The angle between the tangent and chord at the point of contact equals the angle in the alternate segment. 8) A perpendicular from the center bisects a chord.
discriminants.pptx
The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. It defines the discriminant as b^2 - 4ac and explains that:
- If the discriminant is positive, there are 2 solutions
- If the discriminant is 0, there is 1 solution
- If the discriminant is negative, there are no real solutions.
Several examples are provided to demonstrate calculating the discriminant and using it to determine the nature of the roots. The key aspects are to identify the coefficients a, b, and c and then evaluate and compare b^2 - 4ac to determine how many solutions exist.
circles.ppt
The document defines key terms related to circles such as radius, diameter, chord, arc, semicircle, and segment. It explains that a circle is a closed curve where all points are equidistant from the center. A radius connects the center to any point on the circle, while a diameter connects two points on the circle and passes through the center. An arc is part of the circumference between two points on the circle. A semicircle is half of a circle, and a chord connects any two points on the circle. The document also provides examples of circles in daily life, music, and sports.
Lesson 5 Arithmetic Sequences.pdf
This document discusses arithmetic sequences. It defines key terms like sequence, term, and common difference. It explains how to identify if a set of numbers forms an arithmetic sequence based on having a constant difference between terms. Methods are provided for finding the next term, the nth term, and representing the sequence as a linear function. Examples demonstrate how to apply these concepts to solve problems involving arithmetic sequences.
measure of position for group data.pptx
The document discusses measuring positions such as quartiles, deciles, and percentiles in grouped data. It provides steps for calculating each measure of position and interpreting the results. Examples are given to find the second quartile, fifth decile, and 30th percentile in a Scrabble competition. Additionally, the document discusses analyzing energy expenditure data from a Zumba class and grade data from a survey of 10 students.
G-6 Circle.ppt
This document provides definitions and concepts related to circles for a 6th grade mathematics class. It defines key circle terms like diameter, chord, radius, and center. It includes diagrams to illustrate these terms and provides a short quiz for students to test their understanding. The lesson is designed to be 45 minutes and student-centered to teach students about basic circle concepts.
circles.ppt
- A circle is a closed curve in which all points are equidistant from a fixed point called the center.
- Key elements of a circle include the radius, diameter, chord, arc, and semicircle.
- A radius connects the center to any point on the circle, a diameter connects two points on the circle and passes through the center, and a chord connects any two points on the circle.
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Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
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Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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