Polygons
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This document defines key properties and attributes of polygons. It explains that a polygon is a closed figure formed by at least 3 non-intersecting line segments, with sides connecting vertices. A regular polygon is both equilateral and equiangular, while an irregular polygon is not. The interior angles of a convex polygon sum to (n-2)180 degrees, and the exterior angles sum to 360 degrees. Examples are provided to illustrate how to use these properties to calculate angle measures.
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1. The document defines analytic plane quadrants and ordered pairs.
2. An ordered pair (a,b) contains two elements where a is the first component and b is the second. Equality of ordered pairs means the first elements are equal and the second elements are equal.
3. Notations for intervals are defined including closed, open, and half-open intervals using brackets [], parentheses (), and brackets with parentheses [).
Pre geometry
This document provides an overview of basic geometry concepts including:
- Points, lines, planes, and space as the building blocks of geometry.
- Definitions of key terms like points, lines, rays, segments, planes, angles, and different types of angles.
- How to measure angles using a protractor and the units of degrees.
- Relationships between geometric figures like parallel and intersecting lines, coplanar points and lines, and different types of angles.
Solving linear equations in two
1) A first degree equation containing two variables is called a linear equation.
2) There are two methods for graphing linear equations: plotting points that satisfy the equation or using the intercept method of graphing.
3) A set of two or more equations is called a system of equations. There are two methods for solving systems of equations: the elimination method and the substitution method.
Natural numbers
The document defines and describes properties of operations on natural numbers. It defines the set of natural numbers N as the set containing 0, 1, 2, 3, etc. It describes properties of addition, subtraction, multiplication and division of natural numbers such as closure, commutativity, associativity, identity elements and distributivity. Key properties are that natural numbers are closed under addition and multiplication but not subtraction or division.
Logic
This document defines key concepts in logic and propositions:
1) A proposition is a statement that is either true or false, such as "Dogs have six legs" or "2+3=5".
2) The truth value of a proposition is either 1 (true) or 0 (false).
3) Two propositions are equivalent if they have the same truth value. The negation of a proposition, denoted by p', is its opposite truth value.
4) A compound proposition combines two or more simple propositions using logical connectives like "and", "or", or "if/then".
Density
Density is defined as mass per unit volume. The SI unit for density is kg/m3, but g/cm3 is more commonly used. To calculate density, mass is measured using a scale and volume is measured using a ruler or graduated cylinder for irregular objects. Density can be used to differentiate between substances and is a characteristic property of matter. The density of a mixture is calculated by dividing the total mass of the mixture by the total volume.
Mechanics
The document provides formulas and examples for calculating acceleration, velocity, displacement, and time given constant acceleration. It defines acceleration as the rate of change of velocity with time and provides the kinematic equations used to solve constant acceleration problems involving initial velocity, final velocity, acceleration, displacement, and time. Examples include calculating the time, velocity, or displacement of objects accelerating uniformly such as cars, planes, and treadmills.
Divisibility
This document discusses divisibility rules for numbers 2 through 11. It provides examples and explanations of divisibility rules for each number, such as a number is divisible by 2 if its last digit is even, and it is divisible by 3 if the sum of its digits is divisible by 3. The document uses examples such as determining which numbers are divisible by 4 or finding possible digit values for a number to be divisible by 9.
Free fall
All objects accelerate at the same rate when falling near the Earth's surface due to gravity. This constant downward acceleration is denoted as "g" and is about 9.81 m/s^2 on Earth. Free fall acceleration causes objects to fall with increasing downward velocity over time regardless of whether the motion is upward or downward. The document then provides example problems calculating velocity, time, and displacement during free fall.
Prime numbers and factorization
This document defines key concepts related to factors and prime numbers of natural numbers. It explains that factors are numbers that divide a number with no remainder, and that prime numbers only have two factors - 1 and itself. Composite numbers have more than two factors. The prime factors of a number are its factors that are prime numbers. Writing a number as the product of its prime factors is called prime factorization.
Exponents
The document discusses exponents and exponential expressions. It defines exponential expressions as involving a base number being multiplied by itself a certain number of times, called the exponent. It provides examples of exponential expressions and how they are read out loud. It also outlines some properties of exponential expressions, such as all powers of positive numbers being positive and the behavior of even and odd powers of negative numbers. Finally, it discusses like terms in exponential expressions and an identity property involving exponents.
Motion in two dimensions
The document discusses projectile motion, which refers to two-dimensional motion of objects subject to gravity. Examples of projectiles include balls and arrows. Projectiles follow a parabolic trajectory path. The range of a projectile is the maximum horizontal distance it can cover while returning to its original launch height. To analyze projectile motion, the initial velocity vector is resolved into horizontal and vertical components, and kinematic equations are applied to describe motion in each dimension throughout flight.
Force
The document defines force and describes its key properties, including that it can start or stop motion, change speed or direction of motion, or change the shape or size of an object. It also defines different types of forces like gravitational, magnetic, electrostatic, tension, and frictional forces. The document explains that force is measured in Newtons and discusses action-reaction forces.
Radicals
This document discusses working with radicals and square roots. It provides examples of properties of square roots, such as if a2 = b then a is the square root of b where a and b are greater than or equal to 0. The document uses multiple examples to illustrate properties and concepts related to working with radicals and square roots.
Fractions
A fraction represents a part of a whole or group. It consists of a numerator above a fraction bar and a denominator below. Examples include describing parts of a pizza that have been eaten or players on a team with yellow cards. Fractions can be proper if the numerator is smaller than the denominator, improper if the numerator is larger, or mixed numbers combining whole numbers and fractions. Equivalent fractions have the same value, while simplifying reduces fractions to their lowest terms.
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Polygons
- 1. PROPERTIES AND ATTRIBUTES OF POLYGONS
- 2. Definition: • A closed geometric figure which is formed by at least 3 non-intersecting line segments is called a polygon.
- 3. • Each segment that forms a polygon is called side of the polygon. • The common endpoint of two sides is a vertex of the polygon. • A segment that connects any two nonconsecutive vertices is a diagonal .
- 4. • You can name a polygon by the number of its sides.
- 5. • A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. • A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. • If no diagonal contains points in the exterior, then the polygon is convex . • A regular polygon is always convex.
- 7. Polygon Angle Sum Theorem • The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°.
- 8. Examples: • Find the sum of the interior angle measures of a convex octagon. • Find the measure of each interior angle of a regular nonagon. • Find the measure of each interior angle of quadrilateral with angles of x˚, 2x˚, 3x˚ and 4x˚.
- 9. Polygon Exterior Angle Sum Theorem • The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. • The measure of each exterior angle of a regular polygon with n sides is 360˚/n
- 10. Examples: • Find the measure of each exterior angle of a regular hexagon. • Find the measure of each exterior angle of a regular heptagon.