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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). Review • How long is a line? – infinity • Line segment can be measured because – it has two endpoints. • . Congruence • Two segments having the same measure are congruent. A 4 in • Indicated by red slash Z 4 in U B 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 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. X M Z Distance • Calculate Distance on number line – Absolute value of difference between two points. 15 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Distance Coordinate Grid Find the distance between R(5,1) and S(-3,-3) Distance Formula R S R(x1, y1) x1 =5, y1 =1 S(x2, y2) x2 = -3, y2 = -3 Distance Coordinate Grid Find the distance between E(-4,1) and F(3, -1 ) E F Distance Formula Midpoint Exercise Y Z B X C A • 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 What sort of rule did you write for XYZ? How would you write a rule for ABC? Midpoint of Line Segment Number Line Coordinate Plane Endpoints at A and B Endpoints at (x1, y1) and (x2, y2) Midpoint = (A + B) 2 Midpoint = (x1+x2) , (y1+y2) 2 2 Midpoint = (A + B) 2 Calculate Midpoint -12 J 16 M K • Find the Midpoint of JK -12 + 16 2 4 2 =2 • Find the Length of JK |-12 + 16| 4 Midpoint = (x1+x2) , (y1+y2) 2 2 Calculate Midpoint R S Find the midpoint between R(5,1) and S(-3,-3) Midpoint = (5+(-3)) , (1+(-3)) 2 2 2, -2 2 2 (1, -1) 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 Y(-2,2) = (x1+2) , (y1+8) 2 2 -2 = (x1+2) 2 -4 = x1+2 -6 = x1 Solve each problem separately 2= (y1+8) 2 4 = y1+8 -4 = y1 Calculate Measurements 11 + 2x 4x - 5 A C B Find BC if B is the midpoint of AC AB = BC 4x – 5 = 11 + 2x 2x = 16 x=8 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