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Understanding Distance and Midpoints in Geometry: Formulas and Constructions

This lesson covers the essential formulas for calculating distance and midpoints in both two and three dimensions. We will review key concepts, such as the use of absolute value to find distance between points on a number line and how to apply the midpoint formula in coordinate grids. Through practical constructions using a compass, students will learn how to bisect line segments and verify congruence. This hands-on approach emphasizes understanding between points, segment addition, and applying geometric concepts to solve real-world problems.

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Understanding Distance and Midpoints in Geometry: Formulas and Constructions

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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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