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Objective and Standard

1.3 Distance and Midpoints. Objective and Standard. Review distance and midpoint formulas, introduce use of compass to copy and bisect line segments..

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Objective and Standard

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  1. 1.3 Distance and Midpoints Objective and Standard • Review distance and midpoint formulas, introduce use of compass to copy and bisect line segments.. • Check.3.4 Apply the midpoint and distance formulas to points and segments to find midpoints, distances, and missing information in two and three dimensions. • Check.4.3 Solve problems involving betweeness of points and distance between points (including segment addition).

  2. Review • How long is a line? • infinity • Line segment can be measured because • it has two endpoints. • .

  3. Congruence • Two segments having the same measure are congruent. • Indicated by red slash B Z 4 in 4 in U A

  4. Constructions – Equal Lines • What is the precision of the lines you drew? • Use Compass to create a new segment MN that is congruent to TU. • Draw a line on your paper. Draw point M. • Place compass point a T, adjust the setting so the pencil is at point U. • Place point at M, draw and arch that intersects the line and label N • Using compass to create a line segment XZ = XY + YZ

  5. Constructions - Midpoints • Calculate the midpoint of XZ • Use compass to calculate midpoint of XZ, label as M. • Place the compass at X • Adjust settings so the width is greater that ½ XZ. • Draw arcs above and below XZ • Using same compass settings, place compass point at Z and repeat. • Find intersections of the arcs • Using straight edge connect the two points • Label intersection of point as M • Measure segment XM compare against your calculation. M X Z

  6. Distance • Calculate Distance on number line • Absolute value of difference between two points. -8 -7 -6 -2 -10 -9 -5 -4 -3 -1 0 1 2 3 4 5 6 7 8 9 10 15

  7. Distance Coordinate Grid Find the distance between R(5,1) and S(-3,-3) Distance Formula R(x1, y1) x1 =5, y1 =1 S(x2, y2) x2 = -3, y2 = -3 R S

  8. Distance Coordinate Grid Find the distance between E(-4,1) and F(3, -1 ) Distance Formula E F

  9. Midpoint Exercise Y • What are the coordinates of point C? • (2. 5) • What are the lengths of AC and CB? • 3 units each • What are the coordinates of Z? • (-1, 5) • What are the lengths of XZ and ZY? • Between 3 and 4 units each Z B C A X What sort of rule did you write for XYZ? How would you write a rule for ABC?

  10. Midpoint of Line Segment Number Line Endpoints at A and B Midpoint = (A + B) 2 Coordinate Plane Endpoints at (x1, y1) and (x2, y2) Midpoint = (x1+x2) , (y1+y2) 2 2

  11. Midpoint = (A + B) 2 Calculate Midpoint • Find the Midpoint of JK -12 + 16 2 4 2 = 2 -12 16 K J M • Find the Length of JK • |-12 + 16| 4

  12. Midpoint = (x1+x2) , (y1+y2) 2 2 Calculate Midpoint Find the midpoint between R(5,1) and S(-3,-3) Midpoint = (5+(-3)) , (1+(-3)) 2 2 2 , -2 2 2 (1, -1) R S

  13. Calculate Endpoint Find the coordinates of X if Y(-2, 2) is the midpoint of XZ and Z has the coordinates of (2, 8) Let Z = (x2, y2) in the formula (x1+2) , (y1+8) 2 2 Y(-2,2) = Solve each problem separately (y1+8) 2 2 = (x1+2) 2 -2 = 4 = y1+8 -4 = x1+2 -4 = y1 -6 = x1

  14. Calculate Measurements Find BC if B is the midpoint of AC AB = BC 4x – 5 = 11 + 2x 2x = 16 x = 8 11 + 2x C 4x - 5 B A

  15. Assignment – Block Geometry • On a separate sheet of paper – turn in before you leave • Create a line segment AB, of unknown length • Create line EF, of unknown length • Use compass to find midpoint of line EF and label it M • Create third line equivalent to sum of line AB and EM. • Page 31, 12 - 56 every 4th

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