Therefore, x is 3. BC = 2x = 2(3) = Line Segments and Distance. Find each measure. Assume that each figure is not drawn to scale. 1.

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1 Therefore, x is 3. BC = 2x = 2(3) = Line Segments and Distance Find each measure. Assume that each figure is not drawn to scale. 1. CD Thus, BC is CB = 4x 9, BD = 3x + 5, and CD = 17 Here B is between C and D. So, CD = 3.8 in. 2. RS Then x = 3. Use the value of x to find BC. BC = CB = 4(3) 9 = 3 Use the number line to find each measure. So, RS = 2.4 cm. ALGEBRA Find the value of x and BC if B is between C and D. 3. CB = 2x, BD = 4x, and BD = 12 Here B is between C and D. 5. XY The distance between X and Y is 8 So, XY = WZ The distance between W and Z is 9 So, WZ = 9. Therefore, x is 3. BC = 2x = 2(3) = 6. Thus, BC is CB = 4x 9, BD = 3x + 5, and CD = 17 Here B is between C and D. 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. Page 1

2 1-2 Line Segments and Distance The distance between W and Z is 9 So, 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. The distance between A and B is or about A(4, 9), C(9, 0) The distance between A and C is or about B(2, 3), C(9, 0) 7. A(4, 9), B(2, 3) Substitute. The distance between A and B is or about 12.2 The distance between B and C is 8. A(4, 9), C(9, 0) or about ANALYZE RELATIONSHIPS Which two time capsules are the closest to each other? Which are farthest apart? Look at the distances found for each segment AB, AC, and BC. Since BC is the shortest, B and C are the closest to each other. Since AB is the longest distance, A and B are farthest apart. The distance between A and C is or about 10.3 Find each measure. Assume that each figure is not drawn to scale. 11. JL 9. B(2, 3), C(9, 0) Page 2

3 Look at the distances found for each segment AB, AC, and BC. Since BC is the shortest, B and C are 1-2 Line Segments and Distance the closest to each other. Since AB is the longest distance, A and B are farthest apart. Find each measure. Assume that each figure is not drawn to scale. So, PR = 2.1 mm. 15. FG 11. JL So, JL = 1.1 cm. Here, So, FG = GH = HJ = JK. Let each of the lengths be x. 12. EF Therefore, FG = 4.2 cm. 16. WY So, EF = 5.3 in. 13. SV Here, WY = YX. Let WY = YX = x. So, SV = 1.5 in. 14. PR Therefore, WY = 4.4 mm. 17. SENSE-MAKING The stacked bar graph shows the number of canned food items donated by the girls and the boys in a homeroom class over three years. Use the concept of betweenness of points to find the number of cans donated by the boys for each year. Explain your method. So, PR = 2.1 mm. FG 15. esolutions Manual - Powered by Cognero Page 3

4 1-2 Line Segments and Distance Therefore, WY = 4.4 mm. 17. SENSE-MAKING The stacked bar graph shows the number of canned food items donated by the girls and the boys in a homeroom class over three years. Use the concept of betweenness of points to find the number of cans donated by the boys for each year. Explain your method. donated by girls is 20 and that by boys is = 35. For 2011, the bar stops at 75 and the orange colored region stops at 45. So, the number of cans donated by girls is 45 and that by boys is = 30. Find the value of the variable and YZ if Y is between X and Z. 18. XY = 11, YZ = 4c, XZ = 83 Here Y is between X and Z. So, XZ = XY + YZ. The length of the region shaded in orange represents the number of cans donated by girls and the length shaded in blue represents the number of cans donated by boys. For 2009, the bar stops at 45 and the orange colored region stops at 25. So, the number of cans donated by girls is 25 and that by boys is = 20. Similarly, for 2010, the bar stops at 55 and the orange colored region stops at 20. So, the number of cans donated by girls is 20 and that by boys is = 35. For 2011, the bar stops at 75 and the orange colored region stops at 45. So, the number of cans donated by girls is 45 and that by boys is = 30. So, YZ = 4c = 4(18) = XY = 6b, YZ = 8b, XZ = 175 Here Y is between X and Z. Find the value of the variable and YZ if Y is between X and Z. 18. XY = 11, YZ = 4c, XZ = 83 Here Y is between X and Z. So, YZ = 8b = 8(12.5) = XY = 7a, YZ = 5a, XZ = 6a + 24 So, XZ = XY + YZ. So, YZ = 4c = 4(18) = XY = 6b, YZ = 8b, XZ = 175 Here Y is between X and Z. Page 4

5 1-2 Line Segments and Distance So, YZ = 8b = 8(12.5) = XY = 7a, YZ = 5a, XZ = 6a + 24 Here Y is between X and Z. So, YZ = 9d = 9(2) 2 = XY = 4n + 3, YZ = 2n 7, XZ = 22 Here Y is between X and Z. So, YZ = 5a = 5(4) = XY = 11d, YZ = 9d 2, XZ = 5d + 28 Y is between X and Z. 23. XY = 3a 4, YZ = 6a + 2, XZ = 5a + 22 Here Y is between X and Z. So, YZ = 9d = 9(2) 2 = XY = 4n + 3, YZ = 2n 7, XZ = 22 Here Y is between X and Z. So, YZ = 6a = 6(6) + 2 = 38. Use the number line to find each measure. 24. JL Page 5

6 JP = Line Segments and Distance So, YZ = 6a = 6(6) + 2 = 38. Use the number line to find each measure. 29. LN 24. JL LN = 5 Find the distance between each pair of points. JL = JK JK = KP KP = NP The distance between J and K is or about 9.4 NP = JP 31. JP = LN Page 6

7 The distance between J and K is 1-2 Line Segments and Distance The distance between S and T is or about or about The distance between M and L is or about 6.7 The distance between U and V is The distance between S and T is 34. or about 7.6 The distance between X and Y is or about 14.4 Page 7

8 The distance between X and Y is Segments and Distance 1-2 Line or about 14.4 The distance between X and Y is or about P(3, 4), Q(7, 2) 35. The distance between P and Q is or about M ( 3, 8), N( 5, 1) The distance between E and F is or about 14.1 Find the distance between each pair of points. 36. X(1, 4), Y(6, 9) The distance between M and N is or about Y( 4, 9), Z( 5, 3) The distance between X and Y is or about P(3, 4), Q(7, 2) The distance between X and Y is 40. A(2, 4), B(5, 7) or about 6.1 Page 8

9 The distance between X and Y is Segments and Distance 1-2 Line or about A(2, 4), B(5, 7) The distance between C and D is 42. REASONING Vivian is planning to hike to the top of Guadalupe Peak on her family vacation. The coordinates of the peak of the mountain and of the base of the trail are shown in feet. If the trail can be approximated by a straight line, estimate the length of the trail to the nearest tenth of a mile. (Hint: 1 mi = 5280 ft) The distance between A and B is or about 5.4 or about C(5, 1), D(3, 6) Divide by 5280 to convert to miles. The distance between C and D is The length of the trail is about 10 mi. or about REASONING Vivian is planning to hike to the top Determine whether each pair of segments is congruent. 43. of Guadalupe Peak on her family vacation. The coordinates of the peak of the mountain and of the base of the trail are shown in feet. If the trail can be approximated by a straight line, estimate the length of the trail to the nearest tenth of a mile. (Hint: 1 mi = 5280 ft) Here, KJ = HL = 4 in. Therefore, 44. Page 9

10 Here, the lengths of the segments ZY and VW are given to be equal. But the length of UZ is not known. So, the congruency cannot be determined from the information given. Here, KJ = HL = 4 in. 1-2 Line Segments and Distance Therefore, Here, AC = BD = 3 ft. Therefore, Here, MN = RQ = 4x. All segments must have a measure greater than 0. Therefore, for all x > 0, Here, EH = 0.45 cm and FG = 0.5 cm. So, EH FG. Therefore, are not congruent. 46. Here, the lengths of the segments ZY and VW are given to be equal. But the length of UZ is not known. So, the congruency cannot be determined from the information given. Here, SU = 4a + a = 5a and RQ = 2a + 3a = 5a. All segments must have a length greater than 0. Therefore, for all a > 0, 49. TRUSSES A truss is a structure used to support a load over a span, such as a bridge or the roof of a house. List all of the congruent segments in the figure. 47. Here the segments marked with the same symbol are congruent to each other. Here, MN = RQ = 4x. All segments must have a measure greater than 0. Therefore, for all x > 0, 50. CONSTRUCTION For each expression: Page 10 l 48. l construct a segment with the given measure, explain the process you used to construct the segment, and

11 1-2 Line Segments and Distance 50. CONSTRUCTION For each expression: l l l construct a segment with the given measure, explain the process you used to construct the segment, and verify that the segment you constructed has the given measure. a. WZ b. 2(XY) c. 6(WZ) XY a. Sample answer: Step 1: Draw a line and a point on the line. Label the point A. Step 2: Place the compass at point W and adjust the compass setting so that the pencil is at point Z. Step 3: Using that setting, place the compass point at A and draw an arc that intersects the line. Label the second point of intersection B. Since the same arc measure was used to construct from, AB = WZ. b. Sample answer: Step 1: Draw a line and a point on the line. Label the point A. Step 2: Place the compass at point X and adjust the compass setting so that the pencil is at point Y. Step 3: Using that setting, place the compass point at A and draw an arc that intersects the line. From that point of intersection, draw another arc in the same direction that intersects the line. Label the second point of intersection B. Since the same arc measure was used to construct two times, AB = 2(XY). c. Sample answer: Step 1: Draw a line and a point on the line. Label the point A. Step 2: Place the compass at point W and adjust the compass setting so that the pencil is at point Z. Step 3: Using the setting, place the compass at A and draw an arc that intersects the line. From that point of intersection, draw another arc in the same direction that intersects the line. Continue this until you have marked a total of 6 arcs out from A. Step 4: Place the compass at point X and adjust the compass setting so that the pencil is at point Y. Step 5: Using this setting, place the compass at the intersection of the last arc drawn and the line. From that point draw an arc in the opposite direction (towards point A). Label the point of intersection B. Since arc measure WZ was used six times in the same direction and arc measure XY was used to come back in the opposite direction, AB = 6(WZ) XY. 51. BLUEPRINTS Use a ruler to determine at least five pairs of congruent segments with labeled endpoints in the blueprint below. When using the student edition, the lengths, BD = CE = PQ, YZ = JK, PQ = RS, and GK = KL. Therefore, For other forms of media, the answer will vary. Note that you can find congruent segments other than the given too. 52. MODELING Penny and Akiko live in the locations shown on the map below. Since the same arc measure was used to construct two times, AB = 2(XY). c. Sample answer: Step 1: Draw a linebyand a point on the line. Label esolutions Manual - Powered Cognero the point A. Step 2: Place the compass at point W and adjust the compass setting so that the pencil is at point Z. Page 11

12 forms of media, the answer will vary. Note that you can find congruent segments other than the given too. 1-2 Line Segments and Distance 52. MODELING Penny and Akiko live in the locations shown on the map below. 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. The distance between their houses after their move is about 8.2 blocks. 53. MULTI-STEP Coach Willis designs a play that 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? a. The coordinates of the houses are: Penny (4, 6) and Akiko (7, 1) requires the ball to be passed from point A to point E as shown below. The arrows represent quick passes to different members of his team. Randi can throw the ball from under the basket to midcourt, Jen and Mandy can throw the ball half the width of the court, Makayla can throw the ball to the free throw line from under the basket, and Kim can throw the ball farther than Jen. Use the Distance Formula to find the distance between their houses. a. In which position should each girl be? b. Describe your solution process. c. What assumptions did you make? The distance between their houses is about 5.8 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. a. A: Kim, B: Jen or Mandy, C: Jen or Mandy, D: Randi, E: Makayla b. I know that Makayla can throw the shortest distance of all 5 team members and that Randi can throw the farthest. So I chose to place these players at points E and D, respectively. Jen and Mandy can each throw 25 feet (half the width of the court), so I placed them at either point B or point C. Kim will be at point A, and the team will be able to complete the play provided Kim can throw at least 33.3 feet. c. I assumed that the player at point B would be 25 feet from point C. 54. MULTIPLE REPRESENTATIONS Betweenness of points ensures that a line segment may be divided into an infinite number of line segments. Page 12 a. Geometric Use a ruler to draw a line segment 3 centimeters long. Label the endpoints A and D. Draw two more points along the segment and label them B

13 c. I assumed that the player at point B would be 25 feet from point C. MULTIPLE REPRESENTATIONS 54.Line 1-2 Segments and Distance Betweenness of points ensures that a line segment may be divided into an infinite number of line segments. a. Geometric Use a ruler to draw a line segment 3 centimeters long. Label the endpoints A and D. Draw two more points along the segment and label them B and C. Draw a second line segment 6 centimeters long. Label the endpoints K and P. Add four more points along the line and label them L, M, N, and O. b. Tabular Use a ruler to measure the length of the line segment between each of the points you have drawn. Organize the lengths of the segments in and into a table. Include a column in your table to record the sum of these measures. c. Algebraic Write an equation that could be used to find the lengths of and. Compare the lengths determined by your equation to the actual lengths. a. Step 1:Use a ruler to draw a line segment 3 centimeters long Step 2: Label the endpoints A and D. Step 3: Draw two more points along the segment and label them B and C. c. The equation for is AD = AB + BC + CD. is KP = KL + LM + MN + NO The equation for + OP. The lengths of each segment add up to the length of the whole segment. 55. WRITING IN MATH If point B is between points A and C, explain how you can find AC if you know AB and BC. Explain how you can find BC if you know AB and AC. If point B is between points A and C then, AC = AB + BC. So, if you know AB and BC, add AB and BC to find AC. If you know AB and AC, solve the equation to find BC. Therefore, subtract AB from AC to find BC. 56. TOOLS Draw a segment that measures between 2 and 3 inches long. Then sketch a segment, draw a segment congruent to congruent to, and construct a segment. Compare your methods. congruent to Step 4: Draw a second line segment 6 centimeters long. Step 5: Label the endpoints K and P. Step 6: Add four more points along the line and label them L, M, N, and O. b. Using your ruler and the line segments and, measure each section. Then total the lengths and record in the table. c. The equation for is AD = AB + BC + CD. is KP = KL + LM + MN + NO The equation for + OP. The lengths of each segment add up to the length of the whole esolutions Manual segment. - Powered by Cognero 55. WRITING IN MATH If point B is between points and were created using a ruler, while was created using a straightedge and compass and was created without any of these tools.,, and have the same measure, but not only does not have the same length, it isn t even a straight line. Both 57. REASONING Determine whether the statement If point M is between points C and D, then CD is greater than either CM or MD is sometimes, never, or always true. Explain. The statement is always true. If the point M is between the points C and D, then CM + MD = CD. Since measures cannot be negative, CD, whichpage 13 represents the whole, must always be greater than either of the lengths of its parts, CM or MD.

14 was created without any of these tools.,, and have the same measure, but 1-2 Line not Segments and only does notdistance have the same length, it isn t even a straight line. and 57. REASONING Determine whether the statement If point M is between points C and D, then CD is greater than either CM or MD is sometimes, never, or always true. Explain. The statement is always true. If the point M is between the points C and D, then CM + MD = CD. Since measures cannot be negative, CD, which represents the whole, must always be greater than either of the lengths of its parts, CM or MD. 58. CHALLENGE 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? The vertical distance from D to A equals 13 4 = 9 So P is located units to the below point D or 13 3 = 10. So the coordinates of point P are (5, 10). 59. WRITTING IN MATH Explain how the Pythagorean Theorem is used to derive the Distance Formula. The Pythagorean Theorem relates the lengths of the legs of a right triangle to the length of the hypotenuse using the formula c = a + b. If you take the square root of the formula, you get Think of the hypotenuse of the triangle as the distance between the two points, the a value 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 Formula,. Then he draws a line and labels 60. Jonah draws point X on the line. Next, he places the compass point at P and opens the compass so the pencil is at point Q. Using that setting, he places the compass point at X and draws an arc that intersects the line at point Y. By the Segment Addition Postulate, so. We are given that.,, so P is located of the distance from D to A. The horizontal distance from D to A equals 7 1 = 6 So P is located units to the left of point D or 7 2 = 5. The vertical distance from D to A equals 13 4 = 9 So P is located units to the below point D or 13 3 = 10. So the coordinates of point P are (5, 10). 59. WRITTING IN MATH Explain how the Pythagorean Theorem is used to derive the Distance Formula. The Pythagorean Theorem relates the lengths of the legs of a right triangle to the length of the hypotenuse using the formula c = a + b. If you take the square root of the formula, you get Think of the hypotenuse of the triangle as the distance between the two points, the a value as the horizontal distance x2 x1, and the b value as the vertical distance y 2 y 1. Which of the following must be true? I. PQ = XY II. III. PX = QY A I only B II only C I and II only D I, II, and III By using the compass to measure the distance of PQ, XY is than equal in length. Since their lengths are equal, the line segments are congruent. Thus, I. PQ = XY and II. are true. The correct choice is C. 61. Shauntay has a thumbtack that is paper clip shown below. as long as the Page 14

15 By using the compass to measure the distance of PQ, XY is than equal in length. Since their lengths are equal, the line segments are congruent. Thus, I. PQ = XY Segments and II. are true. The correct choice 1-2 Line and Distance is C. 61. Shauntay has a thumbtack that is paper clip shown below. as long as the 22.8 = 4RS RS = 5.7 RU = RS + ST + TU = 3RS RU = 3RS = 3(5.7) = is 7.8 m long. If B lies between A and C and AB is three times the length of, what is AC? A 2.6 m B 5.2 m C 10.4 m D 23.4 m Which of the following is the length of the thumbtack? F 1.2 cm G 10.8 cm H 3.6 cm J 0.33 cm Using the ruler, you can determine the length of the paperclip is about 3.6 cm. Multiply by to find the length of the thumbtack.. Thus the correct choice is F. 62. What is the length of? AB = 7.8 AB = 3BC AB + BC = AC AC = Xavier draws and point Y, which lies between X and Z. If is as long as, which statement correctly describes in terms of? XY + YZ = XZ RV = RS + ST + TU + UV Since all of the segments are congruent, then RV = 4RS = 4RS RS = 5.7 RU = RS + ST + TU = 3RS RU = 3RS = 3(5.7) = is 7.8 m long. If B lies between A and C and AB is three times the length of, what is AC? A 2.6 m B 5.2 m C 10.4 m D 23.4 m AB = 7.8 AB = 3BC is as long as 65. Isaac draws. so that MN = 5.2 cm. Then he draws a point P between M and N so that P is the distance from M to N. Which of the following is the length of? A 1.3 cm B 3.9 cm C 20.8 cm D 15.6 cm of Draw a line segment 5.2 cm. Divide it into 4 equal sections. P is the distance from M, or at the first point to the right of M. Page 15 AB + BC = AC P is the distance away from N.

16 Use the compass to determine the distance from Point A to Point B. Draw Point C on the paper. Draw a ray from Point C. Use the compass to draw and arc that intersects the ray and label it Point D. 1-2 Line Segments and Distance is as long as. 65. Isaac draws so that MN = 5.2 cm. Then he draws a point P between M and N so that P is the distance from M to N. Which of the following is the length of? A 1.3 cm B 3.9 cm C 20.8 cm D 15.6 cm of Draw a line segment 5.2 cm. Divide it into 4 equal sections. P is the distance from M, or at the first point to the right of M. P is the distance away from N. NP = Thus, the correct choice is B. 66. MULTI_STEP Perform the geometric constructions of the following figures using a compass and a straightedge. Then provide an explanation on how each construction was formed. a. Construct a segment. b. Construct a congruent segment. c. Construct a segment that is one half of the original segment. d. Construct a segment that is three times the length of the original segment. c. Create a congruent segment of the original segment using the steps from part b. Then using the compass, open it up wider than half the segment. place pointed part on one endpoint and make an arc above and below the segment. Repeat on the other endpoint making sure the arcs intersect. Use the straightedge to draw a straight line connecting the arcs. The resulting two segments will be congruent and half the length of the original segment. d. Draw a congruent segment to like part b and label the first endpoint as point G and the second endpoint point B. Extend the ray past point B.Then place pointed end on Point B and draw an arc that intersects the ray. Then put the compass with the same settings on the new point created by the segment and the arc intersection. Draw another intersecting arc in the same direction as the previous arc. Label that intersection as point H. a. Draw a Point A on the paper. Then use a straightedge to extend from Point A to Point B. b. Use the compass to determine the distance from Point A to Point B. Draw Point C on the paper. Draw a ray from Point C. Use the compass to draw and arc that intersects the ray and label it Point D. Page 16

Classwork Problems and Homework Packet - Unit 1

Classwork Problems and Homework Packet - Unit 1 Multi-Step Equations DIRECTIONS: Solve each equation. Name NOTE: All answers hav been provided. You are responsible for showing all the work to arrive at the correct answer listed. Date Period 1) 20 =