1-3 Locating Points and Midpoints. Use the number line to find the coordinate of the midpoint of each segment. 18. SOLUTION: E is at 6 and L is at 11.
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1 Use the number line to find the coordinate of the midpoint of each segment Find the coordinates of the midpoint of a segment with the given endpoints 24 J( 112, 34), K( 56, 78) 18 Use the Midpoint Formula E is at 6 and L is at 11 The midpoint of is ( 84, 56) The midpoint is 25 ( 84, 56) 25 esolutions Manual - Powered by Cognero Page 1
2 Find the coordinates of the missing endpoint if B is the midpoint of 28 A(1, 7), B( 3, 1) Let the coordinates of C be (x, y) Then by the Midpoint Formula, 26 Write two equations to find the coordinates of C Use the Midpoint Formula The midpoint of is The coordinates of C are ( 7, 5) C( 7, 5) 32 esolutions Manual - Powered by Cognero Page 2
3 Let the coordinates of A be (x, y) Then by the Midpoint Formula, Write two equations to find the coordinates of A The coordinates of A are esolutions Manual - Powered by Cognero Page 3
4 Suppose M is the midpoint of 34 FM = 5y + 13, MG = 5 3y, FG =? If M is the midpoint, then FM = MG Find the missing measure 36 FM = 8a + 1, FG = 42, a =? If M is the midpoint, then Substitute So, FM = 21 Then y = 1 FM = 5y + 13 = 5( 1) + 13 = 8 MG = 5 3y = 5 3( 1) = 8 FG = FM + MG = = esolutions Manual - Powered by Cognero Page 4
5 ANALYZE RELATIONSHIPS Refer to the number line 41 Find the point X on that is of the distance from A to F 38 Find the point X on that is of the distance from C to F The distance from A to F is 12 units To find the point of the distance from A to F, find The distance from C to F is 9 unit To find the point of the distance from C to F, first find or 18 Then add this to the coordinate of C so point X is at -22 on the number line Add this distance to the coordinate of A to find that point X is at 26 on the number line esolutions Manual - Powered by Cognero Page 5
6 42 Find X on that is the distance from A to B Find the distance between the x-coordinates of A and B Since the ratio of the measure is 1:2, 2JX = XK So, JK = JX + XK = JX + 2JX or 3JX Thus, JX is of JK Multiply the distances by the fractional distance Add this to the x-coordinate of A to determine the x-coordinate of X The x-coordinate of X is 36 Then, find the distance between the y-coordinates of A and B Multiply the distances by the fractional distance Add this to the y-coordinate of A to determine the y-coordinate of X The y-coordinate of X is 22 Thus, point X is located at ( 36, 22) ( 36, 22) 44 Find X on such that the ratio of JX to XK is 1:2 Find the distance between the x-coordinates of J and K Multiply the distances by the fractional distance Add this to the x-coordinate of J to determine the x-coordinate of X The x-coordinate of X is 1 Then, find the distance between the y-coordinates of J and K Multiply the distances by the fractional distance Add this to the y-coordinate of J 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 6
7 48 APPLY MATH Points A and B represent two cities Where should the state place a rest area so it is halfway between cities A and B? 50 GEOMETRY One endpoint of has coordinates ( 3, 5) If the coordinates of the midpoint of are (2, 6), what is the length of? First find the length of Substitute the coordinates for A and B into the midpoint formula The distance from A to the midpoint is 121, thus the distance of the whole segment is (1, 3) 242 esolutions Manual - Powered by Cognero Page 7
8 54 JUSTIFY ARGUMENTS Is the point one third of the way from ( x 1, y the point? Explain Sample answer: Choose some points that lie on horizontal, vertical, and di distance between the first pair of points and the first point and the new po (x 1, y 1 ) (x 2, y 2 ) Distance between f ( 3, 0) (6, 0) (0, 1) (0, 13) (0, 0) (6, 0) (9,12) (0, 0) (0, 0) (12, 9) ( 4, 5) (5, 7) (3, 2) (3, 4) 56 Jamar plots two points, P and Q, on a coordinate plane The midpoint of the points is M( 3, 4) Which of the following could be the points that Jamar plots? A P( 5, 10) and Q(1, 2) B P( 2, 6) and Q( 4, 2) C P( 7, 1) and Q(4, 3) D P( 1, 7) and Q(2, 3) To find the endpoints of Jamar's plots, find the midpoint for each answer choice A This does not match the midpoint ( 3, 4) B This matches the midpoint So, P( 2, 6) and Q( 4, 2) are the endpoints C This does not match the midpoint ( 3, 4) Test each pair of distances Only 2 = and 5 = So when (x way from (x 1, y 1 ) to (x 2, y 2 ) Therefore, the correct answer is sometim Sample answer: sometimes; when the point (x 1, y 1 ) has coordinates (0, D does not match the midpoint ( 3, 4) B 60 Points A, B, C, and D are located on a number line, as shown This Which of the following is the distance from the midpoint of midpoint of? F to the esolutions Manual - Powered by Cognero Page 8
9 G H J Find the midpoint of The midpoint of is 15 Find the midpoint of The midpoint of is 4 The distance between the midpoints is correct choice is J Thus the J esolutions Manual - Powered by Cognero Page 9
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