Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)


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1 Mathematical Sentence  a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement and its negation have opposite truth values* Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true) Disjunction is true when either one part or both parts of the statement is true. (P is true, q is false P\/q is true) Give the original statement p q Converse is formed by interchanging the hypothesis and conclusion. q p Inverse is formed by negating the hypothesis and conclusion. ~p ~q Contrapositive is formed by negating and switching the hypothesis and conclusion. ~q ~P *The conditional and its contrapositive are logically equivalent* Biconditional conjunction of a conditional and its converse (p q) ^ (q p) P q means (p q) ^ (q p) It is true when both p and q have the same truth value. *Tautology is a compound statement that is always true. Opposite of a Tautology is a Contradiction.* Ex. P q [p^(~p\/q)] (p^q) T T T T F T F T T F F T
2 Contradiction a compound statement that is always false Law of Detachment P q (true) p (true) :. q is also true (conclusion) Law of Modus Tollens P q (true) ~q (true) :. ~p (true) OR [(p à q) ^ ~q] à ~P Law of Syllogism p q q r :. p r Law of Detachment (Modus Ponens) p q p :. q Law of Modus Tollens p q ~q :. ~p
3 Line a set of points that extend indefinitely in both directions. Plane a set of points which form a flat surface and extend indefinitely in all directions. Collinear points points that are on the same line. Coplanar points points that are on the same plane. Line segment or segment a set of two points called endpoints and all the points between them. Ray part of a line that starts at one point (called the endpoint) and extends endlessly in one direction Basic Properties/Postulates Reflexive Property a quantity is equal to itself Symmetric Property an equality may be expressed in any other order Transitive Property if quantities are equal to the same quantity, then they are equal to each other Substitution Property a quantity may be substituted for its equal in any expression Midpoint a midpoint divides a segment into 2 congruent segments How to Interpret a Diagram Partition Postulate the whole is equal to the sum of its parts Median of a triangle a segment drawn from ant vertex of a triangle to the MIDPOINT of the opposite side Altitude of a triangle a segment drawn from any vertex of the triangle perpendicular to the line containing the opposite side Circle a set of points in a plane that are a given distance from a given point in that plane Radius a segment drawn from the center of a circle to any point on the circle Can Assume 1. Straight lines and angles. 2. Collinearity of points. 3. Betweenness of points. 4. Relative position of points Can t Assume 1. Right angles 2. Congruent segments. 3. Congruent angles. 4. Relative sizes of segments and angles (not drawn to scale) ALWAYS start off with the Postulate of the Excluded Middle in an indirect Proof Distance between two points is the length of the line segment connecting two points.
4 Distance from a point to a line is the length of the perpendicular segment from the point to the line. The triangle Inequality Postulate  In a triangle, the sum of two side lengths is greater than the length of the third side. Parallelogram a quadrilateral in which both pairs of opposite sides are parallel Each diagonal of a parallelogram splits it into two congruent triangles. Properties: 1. Both pairs of opposite sides are congruent. 2. Both pairs of opposite angles are congruent. 3. Consecutive angles are supplementary. 4. Diagonals bisect each other. Proving a Quadrilateral is a parallelogram 1. Show that both pairs of opposite sides are parallel 2. Show that one pair of opposite sides are both parallel and congruent 3. Show that both pairs of opposite sides are congruent 4. Show that both pairs of opposite angles are congruent 5. Show that diagonals bisect each other 6. An angle is supplementary to both of its consecutive angles Rectangle a parallelogram in which at least one angle is a right angle. Properties 1. Equiangular 2. All the properties of a parallelogram 3. Diagonals are congruent Proving a Quadrilateral is a Rectangle 1. A parallelogram with at least one right angle 2. A parallelogram with congruent diagonals 3. Equiangular
5 Rhombus A parallelogram in which at least two consecutive sides are congruent Properties 1. Equilateral 2. All the properties of a parallelogram 3. Diagonals are perpendicular 4. Each diagonal bisects a pair of opposite angles Proving a Quadrilateral is a Rhombus 1. A parallelogram with at least two adjacent sides congruent. 2. A parallelogram whose diagonals are perpendicular 3. A parallelogram whose diagonals bisect one angle of the parallelogram 4. Equilateral Square a parallelogram that is both a rhombus and a rectangle Properties 1. Equiangular 2. All the properties of a parallelogram 3. Diagonals are congruent 4. Equilateral 5. Diagonals are perpendicular 6. Each diagonal bisects a pair of opposite angles Proving a Quadrilateral is a Square 1. A rhombus with one right angle 2. A rectangle with two congruent adjacent sides Trapezoid a quadrilateral with exactly one pair of parallel sides Isosceles trapezoid a trapezoid in which the nonparallel sides are congruent Properties 1. Lower base angles are congruent 2. Upper base angles are congruent 3. Diagonals are congruent
6 Kite a quadrilateral with two disjoint pairs of consecutive sides are congruent Properties 1. One diagonal is the perpendicular bisector of the other. 2. One of the diagonals bisects a pair of opposite angles. 3. One pair of opposite angles are congruent. Ratio a quotient of two numbers Proportion an equation stating that two or more ratios are equal Mean Proportion a proportion in which the means are equal Ex. ½ = 2/4 = 3/6 Triangle Midsegment Theorem  A segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is ½ the length of the third side. Median (midsegment) of a trapezoid the segment joining the midpoints of the nonparallel sides of a trapezoid Similar polygons (~Polygons) polygons in which 1. All pairs of corresponding angles are congruent 2. The ratios of the lengths of all pairs of corresponding sides are equal Proving triangles similar if there exists a correspondence between the vertices of two triangles such that (AA~ Theorem) Two angles of one triangle are congruent to the corresponding angles of the other then the triangles are similar. (SSS~ Theorem) the ratio of the measures of the corresponding sides are equal then the triangles are similar. (SAS~ Theorem) the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar. Side Splitter Theorem (Triangle Proportionality Theorem) if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally Theorem if an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the given right triangle and to each other
7 Leg 1 Altitude Leg 2 Segment 1 Segment 2 Segment 1 = Altitude Altitude Segment 2 Hypotenuse = Leg 1 Leg 1 Segment 1 Altitude Rule The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse. Leg Rule Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg of the hypotenuse. Angle Side Measures 30 o 60 o 90 o 45 o 45 o 90 o 60 o 45 o 2x X x x 2 90 o 30 o x 3 90 o 45 o x
8 Areas Area of an Equilateral Triangle = (Side) 2 3/4 S Area of a Rhombus = d 1 d 2 /2 d 2 d 1 Scale factor = Ratio of Similitude s 1 /s 2 = P 1 /P 2 (s stands for Side, P stands for Perimeter) Area of a Square = (Side) 2 Area of a Rectangle = (Base)(Height) OR (Length)(Width) Area of a Triangle = ½(Base)(Height) Area of a Trapezoid = ½ (Base 1+Base 2)(Height) OR (Median)(Height) Area of a Ngon = ½ (Apothem)(Perimeter) Median Of a Trapezoid = (Base1+Base2)/2 Radius of a regular polygon is a segment joining the center to any vertex Apothem of a regular polygon is a segment joining the center to the midpoint of any side Lines Distance Formula Distance = (Δx) 2 +(Δy) 2 Midpoint Formula M AB = (x A +X B /2),(Y A +Y B /2) The midpoint is the Average of the two points you re finding the midpoint for. Slope is the rate of change in yvalue for each unit of change in xvalue. Parallel lines have equal slopes. Perpendicular lines have slopes whose product is 1. (Their slopes are negative reciprocals of each other.)
9 Of parallel lines share a point, they are nondistinct. Ways to Write a Line Eqauation Point Slope Formula 1. ax+by=c 2. yy 1 =xx 1 (m) 3. y=mx+b Ways to Solve Linear Equations 1. Graphical Line a 2. Substitution (x,y) Line b 3x+5y=10 x7y=12 Given equations 3x+5y=10 X=12+7y Find point of intersection of these two lines. 3(12+7y)+5y=10 x=12+7y Point of intersection: (5,1) 36+21y+5y=10 x=12+7(1) 26y=26 x=5 Y=1 3. Elimination 10x+3y=2 70x+21y=14 10x+3y=2 Point of Intersection: (1/4,3/2) 6x7y= x21y=36 10(1/4) +3y=2 88x = 225/2+3y= y=9/2 X=1/4 y=3/2
10 Quadratic Equations Quadratic equations can be written in the form y=ax 2 +bx+c Axis of Symmetry Vertical Line given by x OR x= b/2a Vertex; min; max; turning point Vertex found by Substitution of x=b/2a into quadratic equation. To find the y intercept, set x=0 Steps for Graphing Parabolas 1. Find Axis of Symmetry (x=b/2a) 2. Use #1 to find Vertex. 3. Plot 23 points on one side of Axis of Symmetry. (y intercept is an easy one, it s simply c in y=ax 2 +bx+c) 4. Reflect each point across axis of symmetry. 5. Sketch an expanding ushape with arrows. Circle Equation R 2 =(xh) 2 +(yk) 2 At a glance, you can see that (h, k) is the center of the circle. This is in Center Radius form. Completing the Square Ex. x 2 +y 2 +6x8y24 = 0 x 2 +6x+ +y 28y+ =24 To complete the square, halve the linear coefficient for x and y, then add its square to both sides of the equation. (x+3)(x+3) (y4)(y+4) These satisfy the given equation. x 2 +6x+9+y 28y+16= (x+3) 2 + (y4) 2 = 49 Center of the circle = (3, 4) Radius = 7
11 Graphs Solve algebraically, check graphically. 3 Scenarios of Graphs 1. One intersection 2. Two intersections 3. Nothing, no slope Circular Geometry Two or more coplanar circles with the same center are called concentric circles. Two circles are congruent/ equal if they have congruent/equal radii. A point is in the interior of a circle if its distance from the center is less than the radius. A point is in the exterior of a circle if its distance from the center is larger than the radius. A point is on a circle if its distance from the center is equal to the radius. A chord of a circle is a segment joining any 2 points on the circle. A diameter of a circle is a segment (chord) that passes through the center of the circle. The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord. The circumference of a circle is its perimeter. Sector of a circle is a region bounded by two radii and their intercepted arc denoted by 3 letters: center and two endpoints of the arc. Locus Locus a set of points satisfying a given equation Usual Loci a. All points equidistant from a fixed distant, d It is a circle with center d. b. All points equidistant from 2 points A&B The perpendicular bisector of line AB c. All points fixed distance d, from line l Two lines, each parallel to line l, d units to either side of line l d. All points equidistant from 2 parallel lines m & n One line parallel to m & n equidistant from each line
12 e. All points equidistant from 2 intersecting lines. Two lines bisecting angles formed by two given lines f. All points equidistant from sides of an angle. An angle bisector of that angle The perpendicular bisectors of a triangle are concurrent at a point equidistant from the vertices. This point is called the circumcenter. The angle bisectors of a triangle are concurrent at a point equidistant from the sides of the triangle. This point is the incenter. The lines containing the Altitudes of a triangle are concurrent at a point called the orthocenter. The medians of a triangle are concurrent at a point 2/3 of the way from any vertex to its opposite sides. This point is the centroid. Line Symmetry A figure has line symmetry if a line can be drawn such that each side is a mirror image of the other. Transformation change in position, size, or orientation of a figure Line Reflection  transformation that produces a mirror image of a figure on opposite side of the given line r yaxis (x, y) = (x, y) r xaxis (x, y) = (x, y) r y=x (x, y) = (y, x) r y=c (x, y) = (x, 2cy) r x=c (x, y) = (2cx, y) R o (x, y) = (x, y) A figure has point symmetry if it is its own image under a reflection in a point. R o, 90 (x, y) = (y, x) R o, 180 (x, y) = (x, y) R o, 270 (x, y) = (y, x) R y=x (x, y) = (y, x) If you can rotate your figure 180 o about the point in question, and it is still what it used to be, then it has point symmetry. T 3, 2 (x, y) = (x+3, y2)
13 D k (x, y) = (kx, ky) Isometry a transformation that preserves distance Direct isometry is one that preserves order (orientation) Opposite isometry is one that changes order, or orientation, from clockwise to counterclockwise, or viceversa. To reflect an equation in y=x, switch x and y then solve for y. Space Geometry Which of these determine a PLANE? 1 point No Two points No Three collinear points Three noncollinear points A line and a point not on the line Two intersecting lines Two parallel lines No Yes Yes Yes Yes Lateral Area & Total Area Lateral Area of a Cylinder = 2πrh Total Area of a Cylinder = 2πrh + 2πr 2
14 Lateral Area of a Cone = πrl Slant Height (l ) Total Area of a Cone = πrl + πr 2 Total Area of a Sphere = 4πr 2 Volume Volume of a Cylinder = πr 2 h Volume of a Prism = l w h Volume of a Cone = 1/3 πr 2 h Volume of a Regular Pyramid = 1/3 (Area of the base)(height) Volume of a Sphere = 4/3 πr 3 Altitude and Slant Height are different Altitude goes from the tip of the cone to the bottom. Sites you d want to visit. You should also check out the back of your text book, it has all the theorems/postulates. C: GOOD LUCK ON THE REGENTS!! Navida Rukhsha c:
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