* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Slide 1
Survey
Document related concepts
Foundations of mathematics wikipedia , lookup
Location arithmetic wikipedia , lookup
Positional notation wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Large numbers wikipedia , lookup
Non-standard analysis wikipedia , lookup
Hyperreal number wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Line (geometry) wikipedia , lookup
Transcript
Segment Measure and Coordinate Graphing § 2.1 Real Numbers and Number Lines § 2.2 Segments and Properties of Real Numbers § 2.3 Congruent Segments § 2.4 The Coordinate Plane § 2.5 Midpoints 5 Minute-Check 1. Find the next three terms of the sequence 12, 17, 23, 30, … , 38, 47, 57 2. Name the intersection of planes ABC and CDE in the figure. CD D 3. How does a ray differ from a line? A A ray extends in only one direction and has an endpoint. A line extends in two directions. C B 4. Find the perimeter and area of a rectangle with length of 10 centimeters and width of 4 centimeters. P 28 cm A 40 cm 2 E F Real Numbers and Number Lines You will learn to find the distance between two points on a number line. 1) Whole Numbers 2) Natural Numbers 3) Integers 4) Rational Numbers 5) Terminating Decimals 6) Nonterminating Decimals 7) Irrational Numbers 8) Real Numbers 9) Coordinate 10 Origin 11) Measure 12) Absolute Value Real Numbers and Number Lines Numbers that share common properties can be classified or grouped into sets. Different sets of numbers can be shown on number lines. This figure shows the set of _____________ whole numbers . 0 1 2 3 4 5 6 7 8 9 10 The whole numbers include 0 and the natural, or counting numbers. indefinitely The arrow to the right indicate that the whole numbers continue _________. Real Numbers and Number Lines This figure shows the set of _______ integers . -5 -4 -3 negative integers -2 -1 0 1 2 3 4 5 positive integers The integers include zero, the positive integers, and the negative integers. The arrows indicate that the numbers go on forever in both directions. Real Numbers and Number Lines A number line can also show ______________. rational numbers -2 5 3 11 8 -1 1 0 5 3 8 2 3 1 4 3 13 8 A rational number is any number that can be written as a _______, fraction where a and b are integers and b cannot equal ____. zero 2 a b The number line above shows some of the rational numbers between -2 and 2. infinitely many rational numbers between any two integers. In fact, there are _______ Real Numbers and Number Lines decimals Rational numbers can also be represented by ________. 3 0.375 8 2 0.666 . . . 3 0 0 7 terminating or _____________. nonterminating Decimals may be __________ 0.375 0.49 terminating decimals. 0.666 . . . -0.12345 . . . nonterminating decimals. The three periods following the digits in the nonterminating decimals indicate that there are infinitely many digits in the decimal. Real Numbers and Number Lines Some nonterminating decimals have a repeating pattern. 0.17171717 . . . repeats the digits 1 and 7 to the right of the decimal point. A bar over the repeating digits is used to indicate a repeating decimal. 0.171717 . . . 0.17 Each rational number can be expressed as a terminating decimal or a nonterminating decimal with a repeating pattern. Real Numbers and Number Lines Decimals that are nonterminating and do not repeat are called _______________. irrational numbers 6.028716 . . . and 0.101001000 . . . appear to be irrational numbers Real Numbers and Number Lines Real numbers include both rational and irrational numbers. ____________ -2 1.8603 . . . 0 -1 0.8 0.25 3 8 1 0.6 2 1.762 . . . The number line above shows some real numbers between -2 and 2. Postulate 2-1 Number Line Postulate Each real number corresponds to exactly one point on a number line. Each point on a number line corresponds to exactly one real number Real Numbers and Number Lines The number that corresponds to a point on a number line is called the coordinate of the point. _________ On the number line below, 10 __ is the coordinate of point A. The coordinate of point B is __ -4 origin Point C has coordinate 0 and is called the _____. -6 -5 -4 A C B -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x 11 Real Numbers and Number Lines The distance between two points A and B on a number line is found by using the Distance and Ruler Postulates. Postulate 2-2 Distance Postulate For any two points on a line and a given unit of measure, there is a unique positive real number called the measure of the distance between the points. A B measure Postulate 2-3 Ruler Postulate Points on a line are paired with real numbers, and the measure of the distance between two points is the positive difference of the corresponding numbers. A B b measure = a – b a Real Numbers and Number Lines The measure of the distance between B and A is the positive difference 10 – 2, or 8. B -6 -5 -4 -3 -2 -1 0 A x 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 Another way to calculate the measure of the distance is by using ____________. absolute value AB 10 2 BA 2 10 8 8 8 8 Real Numbers and Number Lines Use the number line below to find the following measures. A -3 B -2 1 C -1 5 8 BA 3 3 3 3 1 F 2 0 3 1 CF (1) 2 3 3 2 x Real Numbers and Number Lines Use the number line below to find the following measures. A -3 E D -2 1 -1 1 8 DA 3 3 2 0 3 F 1 2 1 EF 2 3 7 3 5 3 7 3 5 3 x Real Numbers and Number Lines Traveling on I-70, the Manhattan exit is at mile marker 313. The Hays exit is mile marker 154. What is the distance between these two towns? MH 313 154 159 159 miles Real Numbers and Number Lines 5 Minute-Check Find the value or values of the variable that makes each equation true. 1. 3g 63 g = 21 3. 12x 7 67 2 y 2 32 4. 2z 4 3z 6 0 5. If c 4 and d 3, what is the value of the expression 2. 2 5d 3c ? x=5 y=4 or y = -- 4 z = -- 2 6 6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192 Segments and Properties of Real Numbers You will learn to apply the properties of real numbers to the measure of segments. 1) Betweenness 2) Equation 3) Measurement 4) Unit of Measure 5) Precision Segments and Properties of Real Numbers Given three collinear points on a line, one point is always _______ between the other two points. Definition of Betweenness Point R is between points P and Q if and only if R, P, and Q are collinear and _______________. PR + RQ = PQ Q R P PR PQ RQ NOTE: If and only if (iff) means that both the statement and its converse are true. Statements that include this phrase are called biconditionals. Segments and Properties of Real Numbers Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY. Properties of Equality for Real Numbers. Reflexive Property Symmetric Property Transitive Property For any number a, a=a For any numbers a and b, if a = b, then b = a For any numbers a, b, and c, if a = b and b = c then a = c Segments and Properties of Real Numbers Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY. Properties of Equality for Real Numbers. Addition and Subtraction Properties Multiplication and Division Properties Substitution Properties For any numbers a, b, and c, if a = b, then a + c = b + c and a – c = b – c For any numbers a, b, and c, if a = b, then a * c = b * c and a÷c=b÷c For any numbers a and b, if a = b, then a may be replaced by b in any equation. Segments and Properties of Real Numbers If QS = 29 and QT = 52, P Q find ST. S QS + ST = QT QS + ST – QS = QT – QS ST = QT – QS ST = 52 – 29 = 23 T Segments and Properties of Real Numbers If PR = 27 and PT = 73, P Q find RT. R S PR + RT = PT PR + RT – PR = PT – PR RT = PT – PR RT = 73 – 27 = 46 T Segments and Properties of Real Numbers 5 Minute-Check 1. Points X, Y, and Z are collinear. If XY = 32, XZ = 49, and YZ = 81, determine which point is between the other two. X Y Z Refer to the figure below: A Suppose AC = 49 and AB = 14. B 2. Find BC. C BC = AC - AB BC = 49 - 14 BC = 35 3. Suppose D is 5 units to the right of C. What is AD? AD = AC + 5 = 54 Congruent Segments You will learn to identify congruent segments and find the midpoints of segments. In geometry, two segments with the same length are called congruent segments ________ _________ Definition of Congruent Segments Two segments are congruent if and only if they have the same length ________________________ Congruent Segments In the figures at the right, AB is congruent to BC, and PQ is A congruent to RS. The symbol is used to represent congruence. AB BC, and PQ RS. R B C Congruent Segments Use the number line to determine if the statement is True or False. Explain you reasoning. RS TY R -6 -5 S -4 -3 -2 -1 T 0 1 2 Because RS = 4 and TY = 5, 3 4 RS TY So, RS is not congruent to TY, and the statement is false. 5 6 Y 7 8 9 10 11 x Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. 2–1 Congruence of segments is reflexive. AB AB 2–2 Congruence of segments is symmetric. If AB CD, then CD AB 2–3 Congruence of segments is transitive. If AB CD, and CD EF then AB EF Congruent Segments There is a unique point on every segment called the _______. midpoint On the number line below, M is the midpoint of ST. What do you notice about SM and MT? S -6 -5 -4 -3 -2 -1 0 1 2 M 3 SM = MT 4 5 6 T 7 8 9 10 x 11 Congruent Segments A point M is the midpoint of a segment between S and T and SM = MT Definition of Midpoint S M ST if and only if M is T SM = MT The midpoint of a segment separates the segment into two segments of equal _____. length _____ congruent So, by the definition of congruent segments, the two segments are _________. Congruent Segments In the figure, B is the midpoint of AC . Find the value of x. B A 5x - 6 C 2x Check! Since B is the midpoint: Write the equation involving x: Solve for x: AB = BC 5x – 6 = 2x 5x – 2x – 6 = 2x – 2x 3x – 6 + 6 = 0 + 6 AB = 5x – 6 = 5(2) – 6 = 10 – 6 =4 3x = 6 x=2 BC = 2x = 2(2) =4 Congruent Segments To bisect something means to separate it into ___ two congruent parts. midpoint of a segment bisects the segment because it separates the The ________ segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. E Point C bisects AB DC bisects AB D A C EC bisects AB B G Plane GCD bisects AB Congruent Segments 5 Minute-Check 1. DF is bisected at point E, and DF 8. What do you know about the lengths of DE and EF ? The lengths are the same, both are 4. 2. In the figure below, R is the midpoint of QS. Find the value of d. R Q d+4 3. S d + 4 = 3d 4 = 2d 2=d 3d True or False: If AB CD, then CD AB True; segment congruence is symmetric. 4. True or False: If AB BC , then B is the midpoint of AC. False; Points A, B, and C may not be collinear. 5. If a box has 5 red marbles, 5 blue marbles, P ( B ) P (G ) 5 5 10 and 5 green marbles, what is the probability 15 15 15 of selecting either a blue or green marble? The Coordinate Plane You will learn to name and graph ordered pairs on a coordinate plane. In coordinate geometry, grid paper is used to locate points. The plane of the grid is called the coordinate plane. y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 2 3 4 5 The Coordinate Plane The horizontal number line is called the ______. x-axis y 5 4 Quadrant II (–, +) The vertical number line is called the ______. y-axis 3 2 1 Quadrant I (+, +) O x The point of intersection of the two origin axes is called the _____. -5 -4 -3 -2 -1 1 3 4 -1 Quadrant III-2 (–, –) -3 -4 The two axes separate the plane into quadrants four regions called _________. 2 -5 Quadrant IV (+, –) 5 The Coordinate Plane An ordered pair of real numbers, called coordinates of a point, locates a point in the coordinate plane. one point in the Each ordered pair corresponds to EXACTLY ________ coordinate plane. The point in the coordinate plane is called the graph of the ordered pair. graphing the ordered pair. Locating a point on the coordinate plane is called _______ Postulate 2 – 4 Completeness Property for Points in the Plane Each point in a coordinate plane corresponds to exactly one __________________________. ordered pair of real numbers Each ordered pair of real numbers corresponds to exactly one point in the coordinate plane __________________________. The Coordinate Plane Graphing an ordered pair, (point): (x, y) Graph point A at (4, 3) y The first number, 4, is called the x-coordinate ___________. 5 4 (4, 3) 3 It tells the number of units the point lies to left or right of the origin. the __________ 2 (0, 0) 1 x -5 The second number, 3, is called the y-coordinate ___________. It tells the number of units the point lies _____________ above or below the origin. What is the coordinate of the origin? -4 -3 -2 -1 1 -1 -2 -3 -4 -5 2 3 4 5 The Coordinate Plane Graphing an ordered pair, (point): (x, y) Graph point B at (2, –3) y The first number, 2, is called the x-coordinate ___________. 5 4 3 It tells the number of units the point lies to left or right of the origin. the __________ 2 1 x -5 The second number, –3, is called the y-coordinate ___________. It tells the number of units the point lies _____________ above or below the origin. -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 -5 (2, –3) 5 The Coordinate Plane Name the points A, B, C, & D y 5 4 Point A(x, y) = (3, 2) 3 B (–3, 2) 2 Point B(x, y) = (–3, 2) Point C(x, y) = (–3, –2) Point D(x, y) = (3, –2) A (3, 2) 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 C (–3, –2) -3 -4 -5 D (3, –2) The Coordinate Plane On a piece of grid paper draw lines representing the x-axis and the y-axis. y 5 4 Graph : Point A(x, y) = A(2, 4) 3 2 Point B(x, y) = B(2, 0) Point C(x, y) = C(2, –3) 1 x -5 -4 -3 -2 -1 1 2 3 -1 Point D(x, y) = D(2, –5) -2 -3 Consider these questions: -4 -5 x=2 Write a general about pairs thatpoints? have the 1) What do you statement notice about the ordered graphs of these same x-coordinate. They lie on a vertical line. They lie on vertical line that the intersects the x-axis at thepoints? x-coordinate. 2) What doayou notice about x-coordinates of these They are the same number. 4 5 The Coordinate Plane On the same coordinate plane y 5 4 Graph : Point W(x, y) = W(–4, –4) 3 2 Point X(x, y) = X(–2, –4) 1 Point Y(x, y) = Y(0, –4) x -5 -4 -3 -2 -1 1 2 3 4 -1 Point Z(x, y) = Z(3, –4) -2 -3 Consider these questions: y=–4 -4 -5 Write a general about pairs thatpoints? have the 1) What do you statement notice about the ordered graphs of these same y-coordinate. They lie on a horizontal line. They lie on horizontal line that the y-axis at the y-coordinate. 2) What doayou notice about theintersects y-coordinates of these points? They are the same number. 5 The Coordinate Plane If a and b are real numbers, a vertical line contains all points (x, y) such that _____ x=a Theorem 2 – 4 and y=b a horizontal line contains all points (x, y) such that _____ y Graph the lines: x = –3 y=2 5 4 (–3, 2) 3 2 1 Graph the point of intersection of these lines. x -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 2 3 4 5 The Coordinate Plane 5 Minute-Check y 1) Name the coordinates of each point. 4 A = (2, 0) B = (3, –2) 2 A x 0 B -2 -3 -2 -1 1 2 3 4 5 0 2) Graph point C at (0, –4). y x = –4 5 4 3) Graph x = –4. 3 (–4, 2) y=2 2 1 x -5 4) Graph y = 2. -4 -3 -2 -1 1 2 3 -1 -2 -3 5) Graph and label the intersection of x = –4 and y=2 -4 -5 C(0, -4) 4 5 Midpoints You will learn to find the coordinates of the midpoint of a segment. A C B The midpoint of a line segment, AB , is the point C that ______ bisects the segment. C A -7 -6 -5 -4 -3 -2 -1 B 0 1 C = [3 + (-5)] ÷ 2 = (-2) ÷ 2 = -1 2 3 4 5 6 7 Midpoints On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is ab 2 Theorem 2 – 5 A ab 2 B Midpoints Find the midpoint, C(x, y), of a segment on the coordinate plane. Consider the x-coordinate: y x=1 10 It will be (midway) between the lines x = 1 and x = 9 x=9 9 8 A y=7 7 6 C(x, y) Consider the y-coordinate: y 5 4 It will be (midway) between the lines y = 3 and y = 7 y=3 3 B 2 1 x 0 x -1 -2 -2 -1 1 0 2 3 4 5 6 7 8 9 10 Midpoints On a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are x1 x2 y1 y2 , 2 2 y ( x1 , y1 ) Theorem 2 – 6 x1 x2 y1 y2 , 2 2 ( x2 , y2 ) O x Midpoints Find the midpoint, C(x, y), of a segment on the coordinate plane. y x x y y2 C 1 2 , 1 2 2 x=1 10 x=9 9 8 A(1, 7) 73 1 9 C , 2 2 y=7 7 6 C(5, 5) y 5 10 C , 2 10 2 4 y=3 3 B(9, 3) 2 C 5,5 1 x 0 x -1 -2 -2 -1 1 0 2 3 4 5 6 7 8 9 10 Midpoints Graph A(1, 1) and B(7, 9) Draw AB y 10 B(7, 9) Estimate the midpoint of AB. 9 8 7 Check your answer using the midpoint formula. 6 5 x1 x2 y1 y2 C , 2 2 1+7 1+9 C , 2 2 8 10 C , 2 2 C(4, C 5) 4 3 2 1 A(1, 1) x 0 -1 -2 -2 -1 1 0 2 3 4 5 6 7 8 9 10 Midpoints Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B. x-coordinate of B y-coordinate of B y x1 x2 3 2 y1 y2 5 2 10 9 B(-1, 8) 8 7 Replace x1 with 7 and y1 with 2 6 7 x2 3 2 2 y2 5 2 midpoint 4 3 A(7, 2) 2 1 2 y2 10 x 0 -1 -2 -2 Add or subtract to isolate the variable x2 1 C(3, 5) 5 Multiply each side by 2 7 x2 6 B(x, y) is somewhere over there. -1 1 0 y2 8 2 3 4 5 6 7 8 9 10 Midpoints A 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 1 1 2 2 3 4 5 3 6 7 8 9 10 11