1-3 Locating Points and Midpoints

Transcription

1 13 APPLY MATH A business is trying to decide where to build an office The business wants to place the office halfway between city B and city C If city B is at (3, 9) and city C is at (3, 5), find the coordinates of the midpoint Substitute the coordinates for A and B into the midpoint formula 25 X( 24, 14), Y( 6, 68) Use the Midpoint Formula The office should be placed at (3, 2) Find the coordinates of the midpoint of a segment with the given endpoints 23 D( 15, 4), E(2, 10) Use the Midpoint Formula The midpoint of is ( 42, 104) The midpoint of is ( 65, 3) esolutions Manual - Powered by Cognero Page 1

2 Find the coordinates of the missing endpoint if B is the midpoint of 29 C( 5, 4), B( 2, 5) Let A be (x, y) Then by the Midpoint Formula, 27 Use the Midpoint Formula Write two equations to find the coordinates of A The midpoint of is The coordinates of A are (1, 6) esolutions Manual - Powered by Cognero Page 2

3 31 A( 4, 2), B(6, 1) Let the coordinates of C be (x, y) Then by the Midpoint Formula, Suppose M is the midpoint of 35 FM = 3x 4, MG = 5x 26, FG =? If M is the midpoint, then FM = MG Find the missing measure Write two equations to find the coordinates of C The coordinates of C are (16, 4) Then, x = 11 FM = 3x 4 = 3(11) 4 = 29 MG = 5x 26 = 5(11) 26 = 29 FG = FM + MG = = 58 esolutions Manual - Powered by Cognero Page 3

4 37 MG = 7x 15, FG = 33, x =? If M is the midpoint, then 43 Find X on that is the distance from R to S Substitute Find the distance between the x-coordinates of R and S Thus MG = 165 Find x, Multiply the distances by the fractional distance Add this to the x-coordinate of R to determine the x-coordinate of X The x-coordinate of X is 3 Then, find the distance between the y-coordinates of R and S Multiply the distances by the fractional distance Add this to the y-coordinate of R to determine the y-coordinate of X The y-coordinate of X is Thus, point X is located at esolutions Manual - Powered by Cognero Page 4

5 45 Find X on such that the ratio of MX to XN is 2:1 Since the ratio of the measure is 2:1, 1MX = 2XN So, MN = MX + XN = 2XN + XN or 3XN Thus, XN is of MN or Determine the coordinates of the points that satisfy each condition 47 Two points on the x-axis are 10 units from (1, 8) The y-coordinate of the point on the x-axis is 0 So, the point would be of the form (x, 0) Use the Distance Formula to find an expression for the distance between the points (x 0) and (1, 8) and equate it to 10 MX is of MN Find the distance between the x-coordinates of M and K Multiply the distances by the fractional distance Add this to the x-coordinate of M to determine the x-coordinate of X The x-coordinate of X is Then, find the distance between the y-coordinates of M and N There are two possible values for x, 5 and 7 So, the two points are ( 4, 0) and (7, 0) Multiply the distances by the fractional distance Add this to the y-coordinate of M to determine the y-coordinate of X The y-coordinate of X is Thus, point X is located at esolutions Manual - Powered by Cognero Page 5

6 55 MULTI-STEP John wants to center a canvas, which is 8 feet wide, on his living room wall, which is 17 feet wide Where on the wall should John mark the location of the nails, if the canvas requires nails every its length, excluding the edges? Explain your solution process First find the midpoint of the wall Then find the midpoint of the canvas of Since the midpoint of the canvas aligns with the midpoint of the wall, I know that one edge of the canvas will be at 85 4 = 45 feet from the corner of the wall, and the other canvas edge will be at =125 feet from the corner of the wall The canvas requires nails every of its length or every 16 feet, excluding the endpoints So the canvas needs a nail 61 ft, 77 ft, 93 ft, and 109 ft from the corner of the wall esolutions Manual - Powered by Cognero Page 6