1-3 Distance and Midpoints. Use the number line to find each measure.

1-3 Distance and Midpoints
1-3 Distance and Midpoints

1-3 Distance and Midpoints. Use the number line to find each measure.

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1 and Midpoints The distance between W and Zis9So,WZ = 9 Use the number line to find each measure 1XY SOLUTION: TIME CAPSULE Graduating classes have buried time capsules on the campus of East Side High School for over twenty years The points on the diagram show the position of three time capsules Find the distance between each pair of time capsules The distance between X and Yis8So,XY = 8 2WZ SOLUTION: 3A(4, 9), B(2, 3) The distance between W and Zis9So,WZ = 9 TIME CAPSULE Graduating classes have buried time capsules on the campus of East Side High School for over twenty years The points on the diagram show the position of three time capsules Find the distance between each pair of time capsules SOLUTION: The distance between A and B is or about 122 4A(4, 9), C(9, 0) SOLUTION: 3A(4, 9), B(2, 3) SOLUTION: The distance between A and C is 5B(2, 3), C(9, 0) The distance between A and B is or about 122 SOLUTION: or about 103 Page 1

2 So, the midpoint of The distance A and C is andbetween Midpoints or about 103 5B(2, 3), C(9, 0) is whichisthepoint between 6 and 0 8 SOLUTION: SOLUTION: Here, the number line is marked with an interval of 3 The point B is at 0 and D is at 18 Substitute The distance between B and C is or about 76 6CCSSREASONINGWhich two time capsules are the closest to each other? Which are farthest apart? SOLUTION: closest: B and C; farthest: A and C So, the midpoint of between 6 and 12 is9whichisthepoint Find the coordinates of the midpoint of a segment with the given endpoints 9J(5, 3), K(3, 8) SOLUTION: Use the Midpoint Formula Use the number line to find the coordinate of the midpoint of each segment 7 SOLUTION: Here, the number line is marked with an interval of 3 The point A is at 12 and C is at 6 is (4, 55) 10M (7, 1), N(4, 1) SOLUTION: Use the Midpoint Formula So, the midpoint of is whichisthepoint between 6 and 0 8 SOLUTION: Here, the number line is marked with an interval of 3 The point B is at 0 and D is at 18 is (55, 0) Page 2 11Find the coordinates of G if F(1, 35) is the midpoint of andj has coordinates (6, 2)

3 andof Midpoints The midpoint is (4, 55) The coordinates of G are ( 4, 9) 10M (7, 1), N(4, 1) 12ALGEBRA Point M is the midpoint of the value of a in the figure? SOLUTION: Use the Midpoint Formula What is SOLUTION: Use the Midpoint Formula to determine the value of a is (55, 0) 11Find the coordinates of G if F(1, 35) is the midpoint of andj has coordinates (6, 2) SOLUTION: Let the coordinates of G be (x, y) The value of a is 3 Use the number line to find each measure 13JL Write two equations to find the coordinates of G SOLUTION: JL = 5 14JK SOLUTION: JK = 3 15KP The coordinates of G are ( 4, 9) SOLUTION: 12ALGEBRA Point M is the midpoint of the value of a in the figure? What is KP = 9 SOLUTION: Use the Midpoint Formula to determine the value of 16NP SOLUTION: Page 3

4 and Midpoints KP = 9 LN = 5 16NP Find the distance between each pair of points SOLUTION: NP = 2 17JP SOLUTION: 19 SOLUTION: JP = 12 18LN SOLUTION: The distance between J and K is LN = 5 or about 94 Find the distance between each pair of points SOLUTION: SOLUTION: The distance between M and L is The distance between J and K is or about 67 or about 94 Page 4

5 The distance M and L is andbetween Midpoints or about 67 The distance between U and V is SOLUTION: SOLUTION: The distance between X and Y is The distance between S and T is or about or about SOLUTION: SOLUTION: The distance between U and V is 5 The distance between E and F is or about X(1, 2), Y(5, 9) SOLUTION: Page 5

6 The distance E and F is andbetween Midpoints or about X(1, 2), Y(5, 9) The distance between M and N is 28Y( 4, 9), Z( 5, 3) SOLUTION: SOLUTION: The distance between X and Y is or about 81 The distance between X and Y is SOLUTION: SOLUTION: or about 45 The distance between A and B is SOLUTION: SOLUTION: 28Y( 4, 9), Z( 5, 3) SOLUTION: Use Manual the Distance Formula esolutions – Powered by Cognero or about 42 30C(5, 1), D(3, 6) 27M ( 3, 8), N( 5, 1) The distance between M and N is or about 61 29A(2, 4), B(5, 7) 26P(3, 4), Q(7, 2) The distance between P and Q is or about 73 or about 73 The distance between C and D is or about 54 31CCSS REASONING Vivian is planning to hike to the top of Humphreys Peak on her family vacation The coordinates of the peak of the mountain and of the base of the trail are shown If the trail can bepage 6 approximated by a straight line, estimate the length of the trail (Hint: 1 mi = 5280 ft)

7 The distance C and D is andbetween Midpoints a If each square on the grid represents one block and the bottom left corner of the grid is the location of the origin, what is the straight-line distance from Penny s house to Akiko s? b If Penny moves three blocks to the north and Akiko moves 5 blocks to the west, how far apart will they be? or about 54 31CCSS REASONING Vivian is planning to hike to the top of Humphreys Peak on her family vacation The coordinates of the peak of the mountain and of the base of the trail are shown If the trail can be approximated by a straight line, estimate the length of the trail (Hint: 1 mi = 5280 ft) SOLUTION: a The coordinates of the houses are: Penny (4, 6) and Akiko (7, 1) Use the Distance Formula to find the distance between their houses SOLUTION: The distance between their houses is about 58 blocks b If Penny moves 3 blocks to the north, then the coordinates of her house become (4, 9) If Akiko moves 5 blocks to the west, then the coordinates of her house become (2, 1) Use the Distance Formula to find the new distance Divide by 5280 to convert to miles The length of the trail is about 45 mi 32CCSS MODELING Penny and Akiko live in the locations shown on the map below The distance between their houses after their move is about 82 blocks Use the number line to find the coordinate of the midpoint of each segment 33 a If each square on the grid represents one block and the bottom left corner of the grid is the location of the origin, what is the straight-line distance from Penny s house to Akiko s? b If Penny moves three blocks to the north and Akiko moves 5 blocks to the west, how far apart will they be? SOLUTION: H is at 3, and K is at 9 Page 7

8 andbetween Midpoints The distance their houses after their move is about 82 blocks Use the number line to find the coordinate of the midpoint of each segment The midpoint is SOLUTION: F is at 3 and G is at 0 33 SOLUTION: H is at 3, and K is at 9 The midpoint is SOLUTION: F is at 3 and K is at 9 The midpoint is 6 34 SOLUTION: J is at 6 and L is at 11 The midpoint is 3 38 The midpoint is 85 SOLUTION: E is at 6 and L is at SOLUTION: E is at 6 and F is at 3 The midpoint is 25 The midpoint is 45 esolutions Manual – Powered by Cognero 36 SOLUTION: Find the coordinates of the midpoint of a segment with the given endpoints 39C(22, 4), B(15, 7) SOLUTION: Use the Midpoint Formula Page 8

9 and Midpoints The midpoint is 25 Find the coordinates of the midpoint of a segment with the given endpoints 39C(22, 4), B(15, 7) is ( 65, 3) 42V( 2, 5), Z(3, 17) SOLUTION: Use the Midpoint Formula SOLUTION: Use the Midpoint Formula is (05, 6) 43X( 24, 14), Y( 6, 68) is (185, 55) SOLUTION: Use the Midpoint Formula 40W(12, 2), X(7, 9) SOLUTION: Use the Midpoint Formula is ( 42, 104) 44J( 112, 34), K( 56, 78) SOLUTION: Use the Midpoint Formula is (95, 55) 41D( 15, 4), E(2, 10) SOLUTION: Use the Midpoint Formula is ( 84, 56) is ( 65, 3) 42V( 2, 5), Z(3, 17) SOLUTION: Use the Midpoint Formula 45 SOLUTION: Use the Midpoint Formula Page 9

10 and Midpoints is ( 84, 56) is Find the coordinates of the missing endpoint if B is the midpoint of 47C( 5, 4), B( 2, 5) SOLUTION: Let A be (x, y) 45 Write two equations to find the coordinates of A SOLUTION: Use the Midpoint Formula is The coordinates of A are (1, 6) 48A(1, 7), B( 3, 1) 46 SOLUTION: Use the Midpoint Formula SOLUTION: Let the coordinates of C be (x, y) Write two equations to find the coordinates of C is Find the coordinates of the missing endpoint if B is the midpoint of 47C( 5, 4), B( 2, 5) by Cognero esolutions Manual – Powered SOLUTION: Let A be (x, y) Page 10

11 and Midpoints The coordinates of A are (1, 6) 48A(1, 7), B( 3, 1) SOLUTION: Let the coordinates of C be (x, y) The coordinates of C are ( 7, 5) 49A( 4, 2), B(6, 1) SOLUTION: Let the coordinates of C be (x, y) Write two equations to find the coordinates of C Write two equations to find the coordinates of C The coordinates of C are ( 7, 5) 49A( 4, 2), B(6, 1) SOLUTION: Let the coordinates of C be (x, y) Write two equations to find the coordinates of C The coordinates of C are (16, 4) 50C( 6, 2), B( 3, 5) SOLUTION: Let the coordinates of A be (x, y) Write two equations to find the coordinates of A The coordinates of C are (16, 4) Page 11

12 and Midpoints The coordinates of C are (16, 4) The coordinates of A are (0, 8) 50C( 6, 2), B( 3, 5) 51A(4, 025), B( 4, 65) SOLUTION: Let the coordinates of A be (x, y) SOLUTION: Let the coordinates of C be (x, y) Write two equations to find the coordinates of A Write two equations to find the coordinates of C The coordinates of C are ( 12, 1325) The coordinates of A are (0, 8) 52 51A(4, 025), B( 4, 65) SOLUTION: Let the coordinates of C be (x, y) SOLUTION: Let the coordinates of A be (x, y) Write two equations to find the coordinates of C Write two equations to find the coordinates of A Page 12

13 and Midpoints The coordinates of C are ( 12, 1325) The coordinates of A are ALGEBRA Suppose M is the midpoint of Usethegiveninformationtofindthe missing measure or value 53FM = 3x 4, MG = 5x 26, FG =? 52 SOLUTION: Let the coordinates of A be (x, y) SOLUTION: If M is the midpoint, then FM = MG Write two equations to find the coordinates of A Then, x =11 FM = 3x 4 =3(11) 4 =29 MG = 5x 26 =5(11) 26 =29 FG = FM + MG =29+29 =58 The coordinates of A are 54FM = 5y + 13, MG = 5 3y, FG =? ALGEBRA Suppose M is the midpoint of Usethegiveninformationtofindthe missing measure or value 53FM = 3x 4, MG = 5x 26, FG =? SOLUTION: If M is the midpoint, then FM = MG SOLUTION: If M is the midpoint, then FM = MG Then y = 1 Page 13

14 = 5 3( 1) =29 FG = FM + MG =29+29 =58 and Midpoints =8 FG = FM + MG =8+8 =16 54FM = 5y + 13, MG = 5 3y, FG =? 55MG = 7x 15, FG = 33, x =? SOLUTION: If M is the midpoint, then FM = MG SOLUTION: If M is the midpoint, then Substitute Thus MG = 165 Find x, Then y = 1 FM= 5y + 13 =5( 1) + 13 =8 MG = 5 3y = 5 3( 1) =8 FG = FM + MG =8+8 =16 56FM = 8a + 1, FG = 42, a =? SOLUTION: 55MG = 7x 15, FG = 33, x =? SOLUTION: If M is the midpoint, then If M is the midpoint, then Substitute So, FM = 21 Substitute Thus MG = 165 Find x, Page 14 57BASKETBALL The dimensions of a basketball court are shown below Suppose a player throws the ball from a corner to a teammate standing at the

16 in (A2, B2) Use the SQRT function for the square root Use the ^ key to raise to a power of 2 You will need to have several sets parenthesis and Midpoints =SQRT((C2 A2)^2 + (D2 B2)^2) Name the point(s) that satisfy the given condition 61two points on the x-axis that are 10 units from (1, 8) There are two possible values for y, 4 and 10 So, the two points are (0, 4) and (0, 10) 63COORDINATE GEOMETRY Find the coordinates of B if B is halfway between is halfway between andc SOLUTION: 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 SOLUTION: Use the Midpoint Formula to find the coordinates of C There are two possible values for x, 5 and 7 So, the two points are ( 5, 0) and (7, 0) 62two points on the y-axis that are 25 units from ( 24, 3) Use the Midpoint Formula to find the coordinates of B SOLUTION: The x-coordinate of the point on the y-axis is 0 So, the point would be of the form (0, y) Use the Distance Formula to find an expression for the distance between the points (0, y) and ( 24, 3) and equate it to 25 ALGEBRA Determine the value(s) of n 64J(n, n+2), K(3n, n 1), JK = 5 SOLUTION: Use the Distance Formula to find an expression for JK and equate it to 5 There are two possible values for y, 4 and 10 So, the two points are (0, 4) and (0, 10) 63COORDINATE GEOMETRY Find the between is halfway between esolutions Manual – Powered coordinates of B if BbyisCognero halfway Page 16 andc

17 and Midpoints ALGEBRA Determine the value(s) of n 64J(n, n+2), K(3n, n 1), JK = 5 SOLUTION: Use the Distance Formula to find an expression for JK and equate it to 5 a Find the latitude and longitude of the midpoint of the segment between Wilmington and WinstonSalem b Use an atlas or the Internet to find a city near the location of the midpoint c If Winston-Salem is the midpoint of the segment with one endpoint at Wilmington, find the latitude and longitude of the other endpoint d Use an atlas or the Internet to find a city near the location of the other endpoint SOLUTION: a Use the Midpoint Formula 65P(3n, n 7), Q(4n, n + 5), PQ = 13 SOLUTION: Use the Distance Formula to find an expression for PQ and equate it to 13 Round to the nearest tenth = (352, 701) b Sample answer: Fayetteville c Let (x, y) be the location of the other end point Write two equations to find the values of x and y 66GEOGRAPHY Wilmington, North Carolina, is locatedat(343,779 ),whichrepresentsnorth latitude and west longitude Winston-Salem is in the northernpartofthestateat(361,802 ) The latitude and longitude of the other end point are (379, 825) d Sample answer: Prestonburg, Kentucky a Find the latitude and longitude of the midpoint of the segment between Wilmington and WinstonSalem b Use an atlas or the Internet to find a city near the location of the midpoint c If Winston-Salem is the midpoint of the segment with one endpoint at Wilmington, find the latitude and longitude of the other endpoint d Use an atlas or the Internet to find a city near the 67MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between a midpoint of a segment and the midpoint between the endpoint and the midpoint a GEOMETRIC Use a straightedge to draw three PageB 17 different line segments Label the endpoints A and b GEOMETRIC On each line segment, find the midpoint of andlabelitc Then find the midpoint

19 as the horizontal distance x2 x1, and the b value as the vertical distance y 2 y 1 If you substitute, the Pythagorean Theorem becomes the Distance and Midpoints Formula, 69REASONING Is the point one third of the way fro the point? Explain SOLUTION: Sample answer: Choose some points that lie on horiz distance between the first pair of points and the first (x 1, y 1) (x 2, y 2) ( 3, 0) (6, 0) The point P divides the line segment AD in the ratio 2:1 (0, 1) (0, 13) Find the length of AD (0, 0) (6, 0) (9,12) (0, 0) (0, 0) (12, 9) ( 4, 5) (5, 7) (3, 2) (3, 4) Since the ratio of AP to PD is 2 to 1, the ratio of AP to AD is 2 to 3 Find the length of AP Test each pair of distances Only 2 = and 5 = way from (x 1, y 1) to (x 2, y 2) Therefore, the correct 70CHALLENGE Point P is located on the segment between point A(1, 4) and point D(7, 13) The distance from A to P is twice the distance from P to D What are the coordinates of point P? SOLUTION: If point P is on segment AD, then it must satisfy the equation of of Find the slope and equation Write the equation for the line The point P divides the line segment AD in the ratio Use the distance formula and substitution to solve for x if P = (x, y ) Page 19

20 point C on it From C, strike 6 arcs in succession of length AB On the sixth length,performa segment bisector two times to create a and Midpoints Use the distance formula and substitution to solve for x if P = (x, y ) length Label the endpoint D 72WRITING IN MATH Describe a method of finding the midpoint of a segment that has one endpoint at (0, 0) Give an example using your method, and explain why your method works SOLUTION: Sample answer: Divide each coordinate of the endpointthatisnotlocatedattheoriginby2 For example, if the segment has coordinates (0, 0) and ( 10, 6), the midpoint is located at Since P is on, it must be in quadrant I So, x cannot equal 3 Use the equation of the line and x = 5 to determine the value of the y -coordinate for the point P Using the midpoint formula, if the endpoints of the segment are (0, 0) and (a, b), the midpoint is or 73Which of the following best describes the first step in bisecting So, the coordinates of P are (5, 10) 71OPEN ENDED Draw a segment and name it Using only a compass and a straightedge, construct a segment suchthat Explainand A FrompointA, draw equal arcs on usingthe same compass width B FrompointA, draw equal arcs above and below usingacompasswidthof then justify your construction CFrompointA, draw equal arcs above and below SOLUTION: Sample answer: usingacompasswidthgreaterthan DFrompointA, draw equal arcs above and below usingacompasswidthlessthan SOLUTION: Draw Next,drawaconstructionlineandplace point C on it From C, strike 6 arcs in succession of length AB On the sixth length,performa segment bisector two times to create a length Label the endpoint D esolutions Manual – Powered by Cognero IN MATH Describe 72WRITING a method of finding the midpoint of a segment that has one endpoint at (0, 0) Give an example using your The first step in bisecting isfrompointa, draw equal arcs above and below usingacompass width greater than So,thecorrectchoiceisC 74ALGEBRA Beth paid \$7488 for 3 pairs of jeans All 3 pairs of jeans were the same price How much did each pair of jeans cost? F\$2496 Page 20 G \$3744 H \$7488

21 The first step in bisecting isfrompointa, draw equal arcs above and below usingacompass and than MidpointsSo,thecorrectchoiceisC width greater 74ALGEBRA Beth paid \$7488 for 3 pairs of jeans All 3 pairs of jeans were the same price How much did each pair of jeans cost? F\$2496 G \$3744 H \$7488 J \$22464 SOLUTION: Divide \$7488 by = 2496 The correct choice is F 75SAT/ACT If A 04 B 06 C 15 D 16 E2 The correct choice is C 76GRIDDED RESPONSE One endpoint of has coordinates ( 3, 5) If the coordinates of the midpoint of are(2, 6), what is the approximate length of SOLUTION: Let (x, y) be the location of the other end point Then by the Midpoint Formula, Write two equations to find the values of x and y thenx = SOLUTION: 0 Write 1 as 5 If the bases of an equation are equal, then the exponents are equal The coordinates of the other end point are (7, 17) Use the Distance Formula to find the distance between the endpoints The correct choice is C 76GRIDDED RESPONSE One endpoint of has coordinates ( 3, 5) If the coordinates of the midpoint of are(2, 6), what is the approximate length of The length of SOLUTION: Let (x, y) be the location of the other end point Then by the Midpoint Formula, isabout242 Find the length of each object 77Refer to Page 35 SOLUTION: Write two equations to find the values of x and y 78Refer to Page 35 SOLUTION: 38 mm or 38 cm Page 21 Draw and label a figure for each relationship

22 77Refer to Page 35 intersection of the two lines W SOLUTION: and Midpoints 78Refer to Page 35 SOLUTION: 38 mm or 38 cm Draw and label a figure for each relationship lies in plane M and contains point H 79 81TRUCKS A sport-utility vehicle has a maximum load limit of 75 pounds for its roof You want to place a 38-pound cargo carrier and 4 pieces of luggage on top of the roof Write and solve an inequality to find the average allowable weight for each piece of luggage SOLUTION: Draw plane M Place two points in the plane and SOLUTION: Let x be the weight of each piece of luggage Then the combined weight of the 4 pieces of luggage is 4x label them F and G Draw on Place another point The weight of the cargo carrier is 38 pounds The total weight is 4x + 38 The maximum weight is 75 pounds That is, 4x and label it H Solve for x 80Lines r and s intersect at point W SOLUTION: Draw a line and label it r Draw a second line that crosses the first line and label it s Label the point of intersection of the two lines W So, the allowable weight for each piece of luggage is 925 lb or less Solve each equation 82 SOLUTION: Isolate x 81TRUCKS A sport-utility vehicle has a maximum load limit of 75 pounds for its roof You want to place a 38-pound cargo carrier and 4 pieces of luggage on top of the roof Write and solve an inequality to find the average allowable weight for each piece of luggage SOLUTION: Let x be the weight of each piece of luggage Then the combined weight of the 4 pieces of luggage is 4x 83 SOLUTION: The weight of the cargo carrier is 38 pounds The total weight is 4x + 38 The maximum weight is 75 pounds That is, 4x Solve for x 84 So, the allowable weight for each piece of luggage is 925 lb or less SOLUTION: Isolate a Page 22

23 and Midpoints 84 SOLUTION: Isolate a 85 SOLUTION: Isolate k 86 SOLUTION: Isolate z 87 SOLUTION: Isolate n Page 23

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