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Geometry Journal Michelle Habie 9-3 Point, Line, Plane Point: A mark or dot that indicates a location. Ex: Line: A straight collection of dots that go on forever. Ex: Plane: Flat surface that extends forever. Ex: Collinear Points & Coplanar Points: Collinear Points: Points that are in the same line. Ex: Non collinear Points: Points that are not in the same line. Ex: Coplanar Points: Points that are on the same plane. Ex: This is an example of coplanar points that are not collinear. This 3 points are coplanar and collinear. Line, Segment, Ray Line: A straight collection of dots that go on forever. Ex: Segment: A line that has a beginning and an end. Ex: Ray: A line that has a starting point and in one side it keeps on going forever and in the other side, it stops. Ex: The three of them join two points however, some stop and other continues their path. What is an intersection? Intersection: The point where a line crosses the x axis or the y axis. Passing across each other at exactly one point. Exmples: 1. 2. 3. Real Life Intersection Postulate, Axiom, Theorem: Difference: Postulate: A statement that is accepted as true without proof. Axiom: A statement that is accepted as true without proof. Theorem: A statement that has been proven. Ruler Postulate: To measure any segment you use a ruler and subtract the values at the end points. Use the ruler Examples: postulate 1. to find 5 8 the 2. distance from one A B 3. point to the other. Base 1 Base 2 Segment Addition Postulate: If A,B and C are 3 collinear points and B is between A and C then AB+BC=AC. Examples: A B C 1 Home 2 Vista Hermo sa Blvd. 3 CAG Use this postulate to find the distant between 3 or more points on a segment. Distance between 2 points: To find the distance between two points you use the distance formula: √(x2-x1)⌃2+(y2-y1)⌃2 Examples: 1. (2,4) (-1,0) D=√(2+1)⌃2+(4-0)⌃2= D=5 2. (3,-2) (6,-8) D=√(3-6)⌃2+(-2+8)⌃2= D=√45 3. (1,0) (-2,8) D=√(1=2)⌃2+(0-8)⌃2= D=√73 Congruent – Equal: = Two things that have the same value. We have to know the value Comparing Values AB=3.2 ≅ Two things that have equal measure. Might not know what the value is. Comparing Names. --≅-AB CD A, B are congruent and equal. While B, C are congruent but, not equal. Examples: A B C Pythagorean Theorem: The sum of the legs to the squared has to be equal to the hipothenuse to the square. Examples: a2+ b2=c2 A=10 c=16 If a=10 and b=12 Find b Find c: b=√16⌃2-10⌃2 2 + 2 c=√(10 ) (12 ) B=√256-100 = b=√156 C=√244 B=8 c=10 Find a A=√10⌃2-8⌃2 A=√100-64 A=√36=6 Angles and types of angles: An angle is the joining of two rays with a common point called vertex. We measure the angles by the distance from ray to ray, we measure them in degrees. 1. Acute angle (measures 0°-89°) 2. Right angle (measures exactly 90°) 3. Obtuse angle (measures between 91°-179°) 4. Straight angle (measures exactly 180°) The parts of an angle are its legs and the vertex. Legs Interior Legs Vertex Angle addition Postulate: The measurement of two included angles is equal to the measurement of the whole angle that includes both. <CAD+<CAB=<BAD 1. m<CAD= 30° m<CAB=20° Find m<BAD m<BAD=30°+20°=50° B 15° 2. m<BAD= 75° and m<CAD= C Find m< CAB m<CAB=75°-15°=60° A D E F 3. <EIF=32°, <FIG=40°, <GIH=38° Find m<EIH m<EIH=32°+40°+38°=110° G H Midpoint The midpoint is the point that bisects a segment in two congruent parts. You can find a midpoint by measuring or using a straight edge( compass). Examples: 1. F E A B is the midpoint Because lies in the middle of segment A,C. C The apple balances this scale because it represents the midpoint. If EF= 10 and FG= 9, then F is NOT the midpoint of EG. G Angle Bisector: It means to divide an angle into two congruent angles. To construct an angle bisector you use a compass and a ruler to find it. Example 1: Adjacent,Vertical, Linear Pairs of Angles: Adjacent: Have a common vertex sharing a ray with no common interior points. Vertical: Two non adjacent angles form by two intersecting lines. Linear: Adjacent angles with non common sides are opposite rays. Supplementary & Complementary Angles: Supplementary : Any two angles that add up to 180° Complementary : Any two angles that add up to 90° Area and Perimeter of Shapes: A= s⌃ 2 P= 4s Example 1 : s=5in A=(5)⌃2=25in ⌃2 P=4×5=20in Example 2: s=3cm A=(3)⌃2=9cm⌃2 P=4×3=12 cm A= l w P= 2l + 2w Example 1 : l= 2 cm, w= 1 cm A = 2×1=2cm ⌃2 P =2(2)+ 2(1)=6cm Example 2:l= 5 ft, w=3ft A= 5×3=15ft⌃2 P= 2(5)+ 2(3)= 16ft A= ½ l h L= a+b+c Example 1 : l= 5, w= 3, b=4, c=6 A = 5×3÷2= 7.5 u ⌃2 L = 5+4+6= 15u Example 2:l=7, w=4, b=2, c=5 A= 7×4÷2= 14u ⌃2 L=7+2+5= 14u Area and Circumference of a Circle Area= π r ⌃2 Example 1: r= 6 A(3.14)(6)⌃2 A=3.14×36 A=112.04u⌃2 Circumference: 2π r Example 1: r= 8cm c= 2(3.14)(8) c= 50.24cm 5 Step Process: If three cans of soda cost $10.50. How much would seven cans of soda cost? 1. Read it carefully. 2. Understand the problem 3. Make a plan- I definetly need to set up a proportion. 4. 3 cans=7 cans ------ = ------ = 10.50 ×7 10.50 x -----------3 x=$24.50 5. Look Back if the answer makes sense. Rotate Reflect Translate Transformations: Make a copy of a figure in a different position. A transformation can enlarge an object or shrink an object. Examples: Bibliography: • • • • • • • • • • • • http://primaryhomeworkhelp.co.uk/time/pm.gif http://gmat4all.com/diagrams/a1a.jpeg http://www.bced.gov.bc.ca/irp/mathk7/icons/f6.gif http://www.gogeometry.com/heron/angle_bisector.gif http://content.tutorvista.com/maths/content/geometry/lines%20angl es%20triangles/images/img61.gif http://www.freemathhelp.com/images/lessons/angles5.gif http://2000clicks.com/MathHelp/GeometryTheoremsLinearPair.gif http://www.analyzemath.com/Geometry/angle_5.gif http://www.gltech.org/library/April_geometry/supplementary2.gif http://image.wistatutor.com/content/feed/tvcs/translation_0.jpg http://image.wistatutor.com/content/feed/u1856/reflection.GIF http://www.mathsisfun.com/geometry/images/rotation-2d.gif