GEOMETRY Real Numbers Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

GEOMETRY Natural Numbers Whole Numbers Integers Zero Negative Rational Real Numbers Natural Numbers Whole Numbers Integers Zero Negative Integers Rational Numbers Real Numbers Irregular Fraction Irrational Numbers Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Rational Numbers Irrational Numbers GEOMETRY Real Numbers Rational Numbers Rational Numbers = {p/q | p and q are integers and q ≠0} Irrational Numbers Special numbers like pi(∏) or √3 Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Integers Irregular Fraction GEOMETRY Real Numbers Integers Integers = {…-3, -2, -1, 0, 1, 2, 3,…} Irregular Fraction Fractions that has “remainders” like 7/3, 9/5 or 3/2. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Whole Numbers Negative Integers GEOMETRY Real Numbers Whole Numbers Whole Numbers = {0, 1, 2, 3,…} Negative Integers …-3, -2, -1 Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Natural Numbers Zero GEOMETRY Natural Numbers = {1, 2, 3,…} Real Numbers Natural Numbers Natural Numbers = {1, 2, 3,…} Zero Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Real Number Properties GEOMETRY Real Numbers Real Number Properties Commutative a + b = b + a a⋅ b = b ⋅ a Associative (a + b) + c = a + (b + c) (a ⋅ b) ⋅ c = a ⋅(b ⋅ c) Distributive a(b + c) = ab + ac Identity a + 0 = a a ⋅ 1 = a Inverse a + (-a) = 0 a ⋅ 1/a = 1 Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Equality Properties GEOMETRY Addition If a = b, then a + c = b + c Real Numbers Equality Properties Addition If a = b, then a + c = b + c Multiplication If a = b, then a ⋅ c = b ⋅ c Reflexive a = a Symmetric If a = b, then b = a Transitive If a = b and b = c, then a = c Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Equivalence Relation GEOMETRY Real Numbers Definition of Equivalence Relation An equivalence relation is a relation that is reflexive, symmetric, and transitive. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Order of Operations GEOMETRY Do the operations inside the parenthesis. Real Numbers Order of Operations Do the operations inside the parenthesis. PEMDAS: Parenthesis Exponent Multiplication Division Addition Subtraction Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Segment Measure GEOMETRY Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Ruler Postulate Every point of a line can be placed in GEOMETRY Segment Measure Ruler Postulate Every point of a line can be placed in correspondence with a real number. Note: the correspondence is in lowercase as it represents the capital lettered point. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Distance GEOMETRY Segment Measure Definition of Distance The distance between two points A and B is the absolute value of the Difference of their coordinates. Distance between points A and B is Denoted by AB, given by AB = | a - b | Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Segment’s Between GEOMETRY Segment Measure Definition of Segment’s Between A point M is between A and B if AM + MB = AB. The correct notation is A-M-B. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Completeness Postulate GEOMETRY Segment Measure Completeness Postulate Given a ray, AB, and any positive real number r, there is exactly one point C on the ray so that AC = r. A B C r Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Distance Formula For the distance between two points GEOMETRY Segment Measure Distance Formula For the distance between two points on a Cartesian plane, the distance formula is often used: d = √ ((x1 - x2)2 + (y1 - y2)2) Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

GEOMETRY X1 = 3 Y1 = -1 X2 = -7 Y2 = 3 Segment Measure (-7,3) (0,0) (3,-1) Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Segment Bisectors GEOMETRY Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of a Midpoint GEOMETRY Segment Bisectors Definition of a Midpoint The midpoint of AB is M if A-M-B And AM = MB. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Midpoint Theorem If M is the midpoint of AB, then AM = 1/2 AB. GEOMETRY Segment Bisectors Midpoint Theorem If M is the midpoint of AB, then AM = 1/2 AB. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of a Bisector GEOMETRY Segment Bisectors Definition of a Bisector A bisector of a segment is a curve that intersects the segment only at the midpoint. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Congruent Segments GEOMETRY Segment Bisectors Definition of Congruent Segments Congruent segments are segments That have the same length. The symbol ≅is used for congruent segments Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Perimeter and Circumference GEOMETRY Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Perimeter GEOMETRY Perimeter and Circumference Definition of Perimeter Perimeter is the distance around a closed curve. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Theorem 3.2 The perimeter of a regular n-gon with sides of length s is GEOMETRY Perimeter and Circumference Theorem 3.2 The perimeter of a regular n-gon with sides of length s is n (number of sides) x s (sides). Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Circumference GEOMETRY Perimeter and Circumference Definition of Circumference Circumference is the distance around a circle. Circumference = ∏ x d = 2 ∏ r d = diameter r = radius Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Polygons: Inscribed and Circumscribed GEOMETRY Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of an Inscribed Polygon GEOMETRY Polygons: Inscribed and Circumscribed Definition of an Inscribed Polygon An inscribed polygon is a polygon whose vertices are points of a circle. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Circumscribing Circle GEOMETRY Polygons: Inscribed and Circumscribed Definition of Circumscribing Circle A circumscribing circle is a circle that surrounds and contains the vertices of the polygon. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of an Inscribed Angle GEOMETRY Polygons: Inscribed and Circumscribed Definition of an Inscribed Angle An inscribed angle is an angle whose vertex is on a circle and whose sides each contain another point on the circle Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of Circumscribing Polygon GEOMETRY Polygons: Inscribed and Circumscribed Definition of Circumscribing Polygon A polygon circumscribed about a circle is a polygon whose sides each intersect the circle in exactly one point. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of a Tangent Line GEOMETRY Polygons: Inscribed and Circumscribed Definition of a Tangent Line A tangent line (or tangent) is a line in the plane of a circle that intersects the circle in exactly one point. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of a Tangency Point GEOMETRY Polygons: Inscribed and Circumscribed Definition of a Tangency Point The point of tangency is the point at which a tangent line and a circle intersect. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Definition of a Tangent Segment GEOMETRY Polygons: Inscribed and Circumscribed Definition of a Tangent Segment A tangent segment is a segment of a tangent line that contains the point of tangency. Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Tangent Lines, Point of Tangency, and Tangent Segments GEOMETRY Polygons: Inscribed and Circumscribed Tangent Lines, Point of Tangency, and Tangent Segments Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy

Segment Constructions GEOMETRY Joseph Angelo Tiongson. Copyright 2009. Discovery Christian Academy