Chapter 1 Tools of Geometry. Section 1-1 Points, Lines and Planes. Definitions

Transcription

1 Chapter 1 Tools of Geometry Section 1-1 Points, Lines and Planes Definitions Undefined terms - terms that are explained using examples and descriptions Defined terms - terms explained using undefined terms or other defined terms. lso called definitions. Space - a boundless three dimensional set of all points that extends on to infinity

2 Points, Lines and Planes Point - location in space, it has neither size nor shape. Named by a single capital letter Line - n infinite number of points extending on forever in both directions. Named by either a single lowercase script letter or any two points that the line passes through Collinear - lying on the same line Still Points, Lines and Planes Plane - a flat surface that extends on infinitely in all directions. Named by a single capital script letter or by any three non collinear points that lie in the plane Coplanar - lying in the same plane Intersection - the point, line or plane that two or more geometric figures have in common, or share Illustrations Point C l!##"!#" C or C or line C or line C or line l D F E M Plane M or plane DEF or plane EFD or plane FED or plane FDE or plane EDF or plane DFE

3 Example 1 Name the lines and plane!#"!#" or!#"!##"!##" E or D or E E!##"!##"!##" or D or ED or DE C Plane: F D EC, CE, EC, EC CD, DC, DC, CD FE, FED, DEC Example 2 Name the geometric shape modeled by the given object. Patio! patio is a representation of a plane utton on a table button on a table is a representation of a point on a plane Intersections The intersection of two lines is a point The intersection of two planes is a line The intersection of three or more planes is a point It is not possible to draw a true plane, so we used edged shapes to represent planes.

4 Example 3a Draw! and #" label! a ## plane " R that contains the lines and DE which intersect at point P. dd point C on plane R so it is not collinear with the two lines listed before. D P C E R!##" Draw QR on a coordinate plane contains Q(-2, 4) and R(4, -4). dd a new point T that is collinear with these points. Example 3b Q R T -15 Dimensions point occupies no space and has no dimension. line exists in one dimension. Most geometric shapes, like circles and rectangles exist in two dimensions in a plane. n object like a cube or cylinder exists in three dimensions in space.

5 Example 4 How many planes are in the illustration? Name 3 collinear points E F Name the intersection of plane ECD and C What is the! intersection ##"!##" of the lines ED and CE? D C G H Section 1-2 Linear Measure Definitions Line Segment - a portion of a line that has a finite distance. lso known as a segment. Segments are named using endpoints Endpoint(s) - the points used to define the extent(s) of a line segment Unit of measure - the units in which a linear measurement is taken. Inches, feet, etc. If no units of measure are listed, then we use the generic label units to describe the measure.

6 Line Segments is read The measure of line segment it is how we associate a numerical value with the length of the line segment. or Example 1 Find in cm using the given rulers. cm 1 2 cm 1 2 cm cm 1.25 cm 1.2 cm Example 2 Find in inches using the given rulers. in 1 in 1 in in 5 8 in 9 16 in

7 etweenness For any two real numbers a and b there is a real number n that is between a and b such that a < n < b. The same relationship exists with points on a line or segment. Illustration C The point C is between the points and on if C + C = Find XZ Example X Y Z XZ = XY + YZ XZ = 7.3 ( ) + ( 16.2) XZ = 23.5 XZ = 23.5 units

8 Find Example 4 96 in 37 in C C = + C ( 96) = = ( ) Geometric diagrams are not always drawn to scale. 59 = = 59 in Example 5 Find the value of x if T is between S and U, ST = 7x, SU = 45 and TU = 5x - 3. S ST + TU = SU ( 7x) + 5x 3 7x + 5x 3 = 45 12x 3 = 45 12x = 48 x = 4 ( ) = ( 45) 7x T 5x 3 U 45 Congruence and Equality Congruent - Identical in form, coinciding exactly when superimposed. Line segments are said to be congruent segments when their measures are equal. CD D = CD C

9 Example 6 Which segments in the given word in rial font are congruent? TIME Example 6 Which segments in the given word in rial font are congruent? ll the vertical segments are congruent ll the horizontal segments are congruent The two diagonal segments are congruent Constructions Construction - method of creating a figure without the benefit of a measuring tool. Constructions are typically made with only a straightedge, compass and a pencil. These presentations are not the ideal method to illustrate constructions, so if we do constructions they will be done by hand in class.

10 Section 1.2 Extension Precision and ccuracy Precision and ccuracy Precision - refinement in measurement usually represented by the number of digits and describing a cluster of a group of measurements Significant digits - number of digits used in order to express the precision of the measurement ccuracy - a measure of how close the measured value comes to the actual value Error bsolute error - one half of the smallest measurement Relative error - absolute error divided by the expected measure of the object

11 Example 1 Find the absolute error for the measurements: 6.4 cm! ±0.05 cm in ± 1 4 in Significant Digits If we use them to determine precision how do we know what to use? Nonzero digits are significant In whole numbers, zeros are significant if they are between nonzero digits In decimals greater than 1 all digits are significant In decimal numbers less than 1 the first nonzero digit and all digits to the right are significant Example 2 How many significant digits does the number have? ! !

12 Illustration ccurate and Precise ccurate but not Precise Not accurate but Precise Not accurate or Precise Example 3 Calculate the relative error of each measurement: 3.2 miles 0.05 miles! = bout 1.6% 3.2 miles 26 feet 0.5 feet bout 1.9% 26 feet Section 1-3 Distance and Midpoints

13 Distance Distance - the length of the line segment drawn between two points using those two points as its endpoints = x 2 x 1 = x 1 x 2 x1 x2 Example 1 Find the distance between points and on the number line = x 2 x 1 = ( 17) ( 8) = 25 = 25 = x 1 x 2 = ( 8) ( 17) = 25 = 25 The Distance Formula The distance between any two points coordinates (x1, y1) and (x2, y2) in a plane is given by: d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2

14 Example 2 What is the distance between the points (3, -5) and (-2, 4)? d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 d = d = (( 2) ( 3) ) ( 5) 2 + ( 9) 2 d = d = 106 (( ) ( 5) ) 2 Irrational numbers are always a very real possibility. Midpoints Midpoint - a point on a line segment that divides the original segment into two congruent segments M x 1 x 1 + x 2 x 2 2 M = M Example 3 Find the midpoint between points and on the number line x 1 + x 2 2 ( 8) + ( 17) Decimals and fractions are always a very real possibility.

15 The Midpoint Formula The location of the midpoint of any line segment defined by the endpoints (x1, y1) and (x2, y2) in a plane is given by: M x + x 1 2 2, y 1 + y 2 2 Example 4 What is the midpoint between the points (3, -5) and (-2, 4)? M x 1 + x 2 2 M 3 ( ) + ( 2) 2 M 1 2, 1 2, y 1 + y 2 2 ( 5) + ( 4), 2 Example 5 Find the coordinate of D if E(-6, 4) is a midpoint of DF and F(-5, -3). E x + x ( ) 2, y 1 + y 2 2 ( ) E x 1 + 5, y E x 5 1, y E( 6,4) x 1 5 = 6 2 x 1 5 = 12 x 1 = 7 D( 7,11) y 1 3 = 4 2 y 1 3 = 8 y 1 = 11

16 Example 6 Find PR if Q is a midpoint of PR. PQ + QR = PR PQ = QR QR + QR = PR 2QR = PR P 2( 6 3x) = 14x + 2 Q 6 3x R 14x x = 14x + 2 6x = 14x 10 20x = 10 x = 1 2 PR = 14x + 2 PR = PR = PR = 9 isector Segment bisector - any segment, line or plane that intersects a segment at its midpoint. D M C Section 1-4 ngle Measure

17 Definitions Ray - a portion of a line that has a single endpoint and extends out forever in one direction Opposite rays - two rays that share the same endpoint and extend exactly opposite directions to one another. They are collinear ngle - a geometric figure formed by two non collinear rays sharing a common endpoint More Definitions Sides - the two rays that form the angle Vertex - the common endpoint of the two rays where the angle is formed Interior - the area between the two rays that define an angle Exterior - the area outside the two rays that define an angle Illustrations!!" C D CD!!!"!!!" and CE are E opposite rays G F 1 NEVER name an angle by a single point if that point is the vertex of more than one angle. FGH, HGF, G, 1 H

18 Example 1 Name all the angles that have as a vertex. 5, 6, 7, G Name the!!! sides "!!!" of! 5.!!"!!!" G, E or G, F Write another name for 6. DE, DF, ED, FD G E F D 3 4 Measuring ngles Degree - the preferred measuring method for angles. There are 360 degrees in a full circle Right angle - an angle measuring exactly 90 cute angle - an angle measuring less than 90 Obtuse angle - an angle measuring greater than 90 but less than 180 More Measuring ngles Straight angle - an angle measuring exactly 180 Reflex angle - an angle measuring greater than 180 but less than 360

19 Illustrations Right ngle cute ngle Obtuse ngle Straight ngle Reflex ngle Congruent ngles Two angles are said to be congruent if they have equal measures. If m 1 = m 2 then 1 2 C D E F Congruent segments get tic marks, congruent angles get arc marks. ngle isectors ngle bisector - a ray that divides an angle into two congruent angles D!!!" D bisects C C D DC m D = m DC = 1 2 m C

20 Example 3 Find m GH and m HCI if GH HCI and given m GH = 2x + 5 and m HCI = 3x 10 m GH = m HCI 2x + 5 = 3x 10 G H C x + 5 = 10 x = 15 x = 15 K I m GH = 2( 15) + 5 J m GH = E D m GH = 35 m HCI = 35 Section 1-5 ngle Relationships Definitions djacent angles - two angles that lie in the same plane that share a common side and a common vertex and share no interior points Linear pair - adjacent angles with non common sides that are opposite rays Vertical angles - two nonadjacent angles formed by intersecting lines. Vertical angles are congruent

21 More Definitions Complementary angles - two angles with measures that sum up to 90 Supplementary angles - two angles with measures that sum up to 180 These angles do NOT have to be adjacent for this to be true, they don t have to be anywhere near each other. Illustrations djacent ngles D C Linear Pair Vertical ngles More Illustrations G D E F and are complementary and C are supplementary DEG and GEF are supplementary 1 and 2 are complementary 1 2 C

22 Example 1 Name angle pair that satisfied the given condition: Two angles that form a linear pair F G C! Two acute vertical angles E D Example 2 Find the measures of two supplementary angles if the measure of one angle is 6 less than 5 times the measure of the other. m + m = 180 m = 5m 6 5m m = 6 Example 2 Find the measures of two supplementary angles if the measure of one angle is 6 less than 5 times the measure of the other. m + m = 180 m = 5m 6 5m m = 6 6m = 186 m = 31 ( ) 6 m = 5 31 m = 149

23 Perpendicular Lines Perpendicular - forming a right angle. pplies to lines, rays, segments, and planes. D C! C ##"!##" D Example 3!##"!###" Find the values of x and y so KO HM. L M K ( 3y + 6) ( 3x + 6) J 9x O I H 3y + 6 = 90 3y = 84 y = 28 3x x = 90 12x + 6 = 90 12x = 84 x = 7 What We Can ssume Coplanarity Colinearity Points of intersection (not coordinates, just labeled points) etweenness djacent angles Linear pairs Complementary and Supplementary angles

24 What We Can t ssume ngle measure (perpendicular in particular) Congruence (of any figures, but at this point angles and segments specifically) Example 4 Can we assume: m VYT = 90 X Y U T W V S YES TYW and TYU are supplementary YES VYW and TYS are adjacent NO Section 1-6 Two Dimensional Figures

25 Definitions Polygon - a closed future formed by coplanar line segments such that sides that have a common endpoint are not collinear and each side intersects exactly two other sides at their endpoints Vertices - the endpoints of the sides of a polygon. These are also the vertices of the angles formed. Polygons are named according to their vertices in consecutive order. Illustration C Vertices:,,C Sides:, C,C ngles:,, C Name(s): C, C, C C, C, C Polygon C More Definitions Concave - No points or lines containing the sides of the polygon are in the interior of the polygon Convex - Some points or lines containing the sides of the polygon are located in the interior of the polygon

26 More Illustrations Concave Convex More More Definitions n-gon - a generic name for a polygon with n sides Equilateral - having all congruent sides Equiangular - having all congruent angles at the vertices Regular polygon - a polygon equilateral and equiangular at the same time More More Illustrations

27 Commonly Named Polygons Number of Name 3 Triangle or Trilateral or Trigon 4 Quadrilateral or Quadrangle or Tetragon 5 Pentagon 6 Hexagon or Sexagon 7 Heptagon or Septagon 8 Octagon 9 Nonagon or Enneagon 10 Decagon 11 Hendecagon or Undecagon 12 Dodecagon Name that polygon Example 1 Irregular convex quadrilateral Regular concave decagon Not a polygon Perimeter and rea Perimeter - the linear distance around the polygon, the sum of the lengths of the sides of the polygon. Circumference - the linear distance around a circle, equivalent to perimeter rea - the square units required to cover the interior of the figure (circle or polygon)

28 Formulae Triangle Square Rectangle Circle s" d" c" l" r" h" s" s" w" w" d" b" P = b + c + d P = s + s + s + s P = 4s s" l" P = l + w + l + w P = 2l + 2w C = 2πr C = πd = 1 2 bh = s 2 = lw = πr 2 Example 2 Find the perimeter and area of each figure P = 2l + 2w P = P = P = 13.8 cm 4.6 cm ( ) + 2( 2.3) 2.3 cm = lw ( )( 2.3) = 4.6 = 10.6 cm 2 8 in C = 2πr C = 2π ( 8) C = 16π C 50.3 in = πr 2 = π ( 8) 2 = 64π in 2 Example 3 I have 19 feet of rope to mark off an area for my pet goat. Which of the following shapes has a perimeter (or circumference) that would use all or most of the tape?. square with side length 5 ft.. circle with radius of 3 ft. C. right triangle with each leg 6 ft. D. rectangle that is 8 ft by 3 ft.

29 Example 4 Find the perimeter and area of the given shape Example 4 continued P = P = 43 1 = ( 5) ( 5) = 25 2 = 3 3 = 5 ( )( 7) = 21 ( )( 3) = = 1 ( 2 3) ( 4) = = 1 ( 2 5) ( 12) = 30 8 = 97 Section 1-7 Three Dimensional Figures

30 Definitions Polyhedron - solid with all polygon faces Face - every flat surface of a polyhedron, they are polygons Edge - the borders between the faces of a polyhedron, they are line segments Vertex - The point where three or more faces intersect and where 3 or more edges intersect More Definitions Prism - a polyhedron with two congruent faces called bases. Prisms are named by their bases ase - two parallel congruent polygon faces connected by parallelogram faces Pyramid - a polyhedron with one base and three or more triangular faces that intersect at a single vertex. Pyramids are also named for their bases Illustrations

31 Special Focus on Rectangular Prisms rectangular prism is a special case, because every face is a rectangle. While we usually simply view the top and bottom as the bases, there is nothing wrong with considering the right and left sides the bases or the front and back. They all work. More More Definitions Cylinder - a solid with two congruent parallel circular bases connected by a single curved surface Cone - a solid with a single circular base and connected by a curved surface to a single vertex Sphere - all points in space a fixed distance from a single point called the center More Illustrations

32 Example 1 Name the solid, bases, faces, edges and vertices I Solid J F G H Pentagonal Prism ases E D CDE and FGHIJ C Faces!GF!CHG!CDHI!DEJI!EFJ Edges, C,CD, DE,E,FG,GH, HI, IJ, JF, F, G,CH, DI,EJ Vertices,,C,D,E,F,G,H,I,J Special Cases Regular polyhedron - a polyhedron with all of its faces congruent polygons Platonic solid - 5 special types of regular polyhedra used extensively by Plato Platonic Solids Hexahedron Dodecahedron Tetrahedron Octahedron Icosahedron Platonic solids are commonly used in polyhedral dice, used to play games like Dungeons and Dragons.

33 Surface rea and Volume Surface area - a two dimensional measurement of the surface of a solid object Volume - a three dimensional measurement of the space occupied by a solid Remember, surface area is measured in square units like ft 2 and volume is measured in cubic units like cm 3. Formulae Prism Regular Pyramid Cylinder Cone Sphere S = Ph + 2 V = h h h l S = 1 2 Pl + V = 1 3 h r h r S = πrl + πr 2 2 S = 2πrh + 2πr V = πr 2 h V = 1 3 πr 2 h h l r S = 4πr 2 V = 4 3 πr3 S = Total Surface rea, V = Volume, h = height, P = Perimeter of the base, = area of the base, r = radius, l = Slant height Example 2 Find the total surface area and volume of a square pyramid with height of 4 ft, slant height of 5 ft and base side length of 6 ft S = 1 2 Pl + ( )( 5) + ( 6) ( 6) S = S = V = 3 h S = 96 ft 2 V = ( )( 4) V = 48 ft 3

34 Example 3 You have a circular pool that is 20 feet across and 4 feet deep. How much water does it take to fill the pool? 10 4 V = πr 2 h V = π ( 10) 2 ( 4) V = 400π V ft 3 Supplemental Information Naming Polygons and Polyhedra Greek Prefixes Up to 20 Number Prefix 3 tri 4 tetra 5 penta 6 hexa (or sexa) 7 hepta (or septa) 8 octa 9 nona or (or ennea) 10 deca 11 hendeca (or undeca) Number Prefix 12 dodeca 13 triskadeca 14 tetradeca 15 pentadeca 16 hexadeca (or sexadeca) 17 heptadeca (or septadeca) 18 octadeca 19 enneadeca (or nonadeca) 20 icosa ny polygon with 3 to 20 sides is simply the prefix with the term gon at the end

35 Greek Prefixes Over 20 Tens Digit Prefix 20 icosi 30 triaconta 40 tetraconta 50 pentaconta 60 hexaconta 70 heptaconta 80 octaconta 90 enneaconta Ones Digit Prefix 1 hena 2 di 3 tri 4 tetra 5 penta 6 hexa 7 hepta 8 octa 9 ennea Construction Construction of terms from 21 to 99 (technically 3 to 99) follows a particular pattern so we will use a 57 sided figure as an example. Tens Digit and Ones Digit gon pentaconta kai hepta gon 57 sided figure is called a pentacontakaiheptagon Greek Prefixes Over 99 Hundreds Digit Prefix 100 henahecta 200 dihecta 300 trihecta 400 tetrahecta 500 pentahecta 600 hexahecta 700 heptahecta 800 octahecta 900 enneahecta

36 Construction again The term kai is not used between the hundreds and tens digits. Let s try 736 sides. Hundreds Digit Tens Digit and Ones Digit gon heptahecta triaconta kai hexa gon 736 sided figure is called a heptahectatriacontakaihexagon Polyhedra? Everything we just discussed for polygons works equally well with polyhedra as well. Simply add the phrase hedron instead of gon ad the end, and you have the name for your polyhedron.