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Tools of Geometry

Learn how to find the distance and midpoint between two points on a coordinate plane, graph lines, write equations of lines, and identify if lines are parallel or perpendicular. Inductive reasoning and basic terms of geometry are also covered.

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Tools of Geometry

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  1. Tools of Geometry

  2. 1-8 The Coordinate Plane Objective: Find the distance and midpoint between two points Why? Map

  3. Formulas Midpoint ((X1+X2)/2 , (Y1+Y2)/2 ) Distance = ((X1-X2) 2 + (Y1-Y2)2)

  4. Example 1 • Find the midpoint of A(13,8) and B(-6,4). • C is the midpoint of AB. C(6,1) and A(7,10) find the coordinate of point B.

  5. Example 2 Find the distance between A(10,14) and B(4,-2). Round to the nearest tenth. 17.1 What did you learn? What is still confusing?

  6. 3-6 Lines in the Coordinate Plane Objective: To graph lines given their equations To write equations of lines

  7. Linear Equation Forms • Point-slope form (y1 – y2) = m (x1 –x2) • Slope-intercept form Y = mx + b • Standard form Ax + By = C m= slope b = y-intercept A is always positive. No fractions

  8. Formulas Slope = m = y1 – y2 x1 – x2

  9. Identifying the slope and y-intercept of a line. • Y = 2x + 5 m = 2 b = 5 • 3x + 6y = 24 -3x -3x 6y = 24 – 3x 6 6 6 y = 4 – 1/2x m = -1/2 b = 4 Start at 4 on the y axis Down 1 right 2 Graph the line.

  10. Writing equation to a line given a point and the slope. A(3,-2) m = 4 Y = mx + b -2 = 4(3) + b -2 = 12 + b -12 -12 -14 = b Point slope form y +2 = 4( x – 3) or Slope intercept form y = 4x – 14 or Standard form 4x – y = 14

  11. Writing an equation to a line given two points. A(0,4) B(5,-1) m = 4--1 = 5 = -1 0-5 -5 Y = mx + b 4 = -1(0) + b 4 = 0 + b 4 = b Slope intercept form y = -x + 4 Or in standard form X + y = 4 Or in point slope form y + 1 = -( x - 5)

  12. Find the x- and y-intercepts of the line 5x + 3y = 15. Set x = 0 to find y intercept 5(0) + 3y = 15 3y = 15 Y = 5 (0,15) Set y = 0 to find the x-intercept. 5x + 3(0) = 15 5x = 15 X = 3 (3,0)

  13. Write a horizontal and vertical equation for A(2,-5) • Horizontal Equation y = -5 • Try(0,4) • Vertical Equation X = 2

  14. Try. • What is the slope and the y-intercept of the line 2x + 3y = 12 M = -2/3 and b = 4 • Write an equation to the line with the point (2,3) and the slope 1/2. y – 3 = ½(x – 2) or y = ½x +2 or x-2y = -4 • Write an equation to AB given the A(3,9) and B(-2,4) in standard form. x + y = 6

  15. 3-7 Parallel and Perpendicular Lines in the Coordinate Plane Objective: To identify if two lines are parallel, perpendicular, or intersecting. To write equations of lines that are parallel and perpendicular to a line.

  16. How to Tell the Relationships of linear equations Parallel – same slopes Perpendicular – negative reciprocal slopes i.e. -1/2 and 2 5/8 and -8/5

  17. Writing equations that are parallel to a line through a given point 2x + 3y = 12 (18, 8) Find the slope. Write equation with point (18,8) Y – 8 = -2/3(x – 18) point slope form Y – 8 = -2/3x + 12 Y = -2/3x + 20 -- slope intercept form -2/3x + y = 20 Multiply everything by -3 2x -3y = -60 standard form 2x + 3y = 12 -2x -2x 3y = 12 – 2x 3 3 3 y = 4 – 2/3x M = -2/3

  18. Writing equation of line that are perpendicular through and given point Y = 5x + 5 A(-2,4) M = 5 m = -1/5 Y – 4 = -1/5(x + 2) – point-slope form Y – 4 = -1/5x - 2/5 Y = -1/5x +18/5 – slope-intercept form 5y = -x + 18 X + 5y = 18 -- standard form

  19. Determine if the lines are parallel, perpendicular or neither. Perpendicular L1 m = -1/1 = -1 L2 m = 1/1 = 1 perpendicular 3x + 6y = 30 6y = -3x + 30 Y = -1/2x +5 m = -1/2 4y + 2x = 9 4y = -2x + 9 y = -1/2x +9/4 m = -1/2 parallel

  20. 1-1Patterns and Inductive Reasoning Objective: to use inductive reasoning to make conjectures Why? Predict Sales, patterns in nature,…

  21. Vocabulary 1-1 • Inductive reasoning – reasoning based on patterns you observe • Conjecture- a conclusion you reach using inductive reasoning • Counter example – an example that proves a conjecture false

  22. Example 1 Write the next two terms in the sequence. • 384, 192, 96, 48… • M,T,W,T,… TRY: Inductive reasoning & conjecture

  23. Example 2 Make a conjecture about the tenth triangular numbers 1(1+1)/2 3(3+1)/2 2(2+1)/2 4(4+1)/2

  24. Try Make a conjecture about the sum of the cubes of the first 25 counting numbers. 13 =1 = 12 13+ 23 = 9 = (1+ 2)2 13+ 23+33 = 36 = (1+ 2+3)2 13+ 23+ 33+43 = 100 =(1+ 2+3+4)2

  25. Example 3 Find a counter example for each conjecture. • The cube of any number is greater than the original number. • A number is always greater than its reciprocal. Try: Alana makes a conjecture about slicing pizza. She say that if you use only straight cuts, the number of pieces will be twice the number of cuts. What did you learn today? What is still confusing? Class work / homework

  26. 1-3 Points, Lines, and Planes Objective: Understand basic terms of geometry. Why? These are the building block of geometry.

  27. Vocabulary 1-3 Point- location, has no size. Space- set of all points (a geometric figure). Line-a series of points that extends in two opposite directions without end. Collinear points- points that lie on the same line. Plane- flat surface that has no thickness. Coplanar- points and lines on the same plane. Postulate(Axiom)- a ccepted statement of fact

  28. Key Concepts 3) What is created at the intersection of two planes intersect? If two planes intersect, then they intersect in exactly one line. 4)What can you create through three non-collinear points? Through any three non-collinear points there is exactly one plane. 1) How many lines can be made through two points? Through any two points there is exactly one line. 2) If two lines intersect, how many points are formed? If two lines intersect, then they intersect at exactly one point.

  29. How to Name It Point --- A, B, C Line ---line m, AB Plane--- ABC or ABCD --- at least 3 CAPTIAL letters are used D A B C

  30. 1-4 Segments, Rays, Parallel Lines and Planes Objective: 1) Identify segments and rays 2) Recognize parallel lines Why? To identify compass directions, building blocks to geometry

  31. Vocabulary 1-4 Segment- part of a line with two endpoints. Ray- part of a line with one end point. Opposite rays-two collinear rays with the same endpoint.(forms a line) Parallel lines- coplanar lines that do not intersect. || Skew lines- noncoplanar lines tht do not intersect. Parallel planes- planes that do not intersect

  32. Example 1 • Name three collinear points. • Name two different planes that contain the point A and E. • Name the intersection of plane BCD and plane HEA. • How many planes contains the points C, E, and H? • Write a counter example. Two lines in the same plane always intersect. A B C D H G E F A B C D H G E F T

  33. Example 2 Describe the points are collinear, noncollinear, coplanar or noncoplanar. • A,B • F,A,E c)B,I,L,O What is one thing new you learned today? What is still confusing?

  34. How to Name It? Segment AB Ray BC Is AB and BA the same ray? C D A B

  35. Example 1 A B a) Name segments that form a square. b) Name a pair of opposite rays. c) How can you tell if a pair of lines are skew? d) Name a pair of parallel planes e) Name a pair of skew line to AB How are parallel and skew lines alike and different? What is still confusing?

  36. 1-5 Measuring Segments Objective: Find the lengths of segments Why? Measure distance

  37. Key Concepts Ruler Postulate -- |a – b| Segment addition B C A 7 9 3

  38. Vocabulary 1-5 Midpoint – a point that divides the segment in half. Congruent segments - segments of equal length.

  39. Example 1– figure not drawn to scale A B C 1) If AB = 9 and AC = 30, then BC = _____ 2) B is the midpoint of AC. If AB = 5x+9 and BC = 8x – 36, find AB,BC and AC. 3) If AC = 25 and AB = 2x – 6 and BC = x +7, then find x, AC and BC. What did you learn today? What is still confusing? Class work / homework

  40. 1-6 Measuring Angles Objective: find measure of angle and identify special angle pairs Why? Precision flying, rollercoasters,…

  41. How to Name an Angle Angle parts: Naming angles: 1 , ABD ,  DBA 2, DBC, CBD Ray Ray

  42. Vocabulary 1-6 Acute – 0⁰<x<90⁰ Right – 90⁰ Obtuse – 90 ⁰ <x<180 ⁰ Straight 180 ⁰ Congruent angles – angles with the same measure

  43. Vocabulary 1-6 continued Vertical angles – hint bow tie Adjacent angles – common vertex and side Complementary angles are two angles whose measures have the sum of 90 Supplementary angles are two angles whose measures have the sum of 180

  44. Key Concept Angle Addition- sum of two small adjacent angles equals a bigger angle If point C lies in the interior of ABD, then mABC + mCBD = mABD.

  45. EXAMPLE 1 Classify angle 1 and 2. Name  1 two different ways. 2 1

  46. Example 2 • Find mPQR if PQS = 25 and SQR = 35. • Find mPQS if mPQR = 6x-12 and mSQR = 2x-2

  47. Example 3 • Name an angle supplementary to ADC. • Name pair of vertical angles. • Name an angle complementary to CDE A B C D E G F What did you learn today? What is still confusing?

  48. 1-7 Basic Constructions Objective: Use geometry SketchPad to construct segments and angles and bisect angle and segments

  49. Vocabulary 1-7 Perpendicular line – two lines that intersect to form right angles. | Perpendicular bisector – a line, segment or ray that is perpendicular to a segment at its midpoint.

  50. Vocabulary 1-7 continued Angle bisector -- a ray that divides an angle into two congruent coplanar angles What did you learn today? What is still confusing? Class work / homework

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