Geometric Constructions

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1 Geometric onstructions (1) opying a segment (a) Using your compass, place the pointer at Point and extend it until reaches Point. Your compass now has the measure of. (b) Place your pointer at, and then create the arc using your compass. The intersection is the same radii, thus the same distance as. You have copied the length. ' ' 1. Given line segment : a) opy b) onstruct a line segment whose measure is twice 2. Given line segment D: a) opy D b) onstruct a line segment that is there times D. c) onstruct a line segment that is equal to + D 1

2 3. Given, D,& EF. Use the copy segment construction to create the new lengths listed below. E D F a) 3 b) D + EF c) 2D + D d) EF D 4. a) onstruct an equilateral triangle using : b) onstruct an equilateral triangle using D 2

3 c) onstruct an equilateral triangle using EF d) construct a scalene triangle using, D and EF e) onstruct a Isosceles triangle using D as the two legs and as the base: 3

4 1. onstruct an Isosceles triangle use as the legs and D as the base. 2. onstruct a triangle use, D, and EF as the 3 sides. 4

5 3. onstruct an Isosceles triangle use as the legs and D as the base. 4. onstruct an Isosceles triangle use as the legs and D as the base. 5

6 (2) isect a segment (a) Given (b) Place your pointer at, extend your compass so that the distance exceeds half way. reate an arc. (c) Without changing your compass measurement, place your point at and create the same arc. The two arcs will intersect. Label those points and D. D (d) Place your straightedge on the paper so that it forms D. The intersection of D and is the bisector of. M D 1. isect line segment and D a) b) 6

7 2. isect line segment and D a) c) 3. onstruct a line segment that is 1 and half times D: 4. onstruct a line segment that is 2 and half times : 7

8 5. Given & D. Use the midpoint construction to find the midpoint of & D D 6. Use your midpoint construction to determine the exact length of 1 4 EF E F 8

9 median is a line segment that connects the midpoint of one side of a triangle and the opposite vertex. 1. onstruct a median to 2. onstruct a median to 3. onstruct a median to 4. onstruct a median to 9

10 5. onstruct a median to 6. onstruct a median to 7. onstruct a median to 8. onstruct a median to 10

11 (3) opy an angle (a) Given an angle and a ray. (b) reate an arc of any size, such that it intersects both rays of the angle. Label those points and. (c) reate the same arc by placing your pointer at. The intersection with the ray is. ' ' ' ' (d) Place your compass at point and measure the distance from to. Use that distance to make an arc from. The intersection of the two arcs is. (e) Draw the ray ' ' (f) The angle has been copied. o ' ' ' ' ' ' ' ' o ' 1. opy 11

12 2. opy 3. opy 4. onstruct and angle twice 12

13 5. onstruct and angle twice 6. opy 7. onstruct and angle three times D 13

14 8. Given. Make a copy of, ' ' '. ' 9. Given DEF. Make a copy of DEF, DEF ' ' '. D F E E' 14

15 10. Given MN, construct 2.5 MN M N 11. Given GH, construct 1.75 GH G H 12. Given Δ, construct a copy of it, Δ. ' 15

16 13. Given V -- perform the midpoint construction. This time labeling the two intersection found to be H and K. Draw in VH, VK, H,& K. lso draw HK. V Why is VH = VK? Why is H = K? Why is VH = VK = H = K? What is the most specific name for the quadrilateral VHK? Will this specific quadrilateral be formed every time using this construction? Yes or No Why or why not Label the intersection of HK and V is point M. What is true about VM and M? What is true about HM and KM? What is the measure of the angle formed at the intersection of HK and V? 16

17 1. onstruct a Triangle using the following angle and 2 sides 3. onstruct a Triangle using the following angles and side. 17

18 3. onstruct a Triangle using the following angles and side. 2. onstruct a Triangle using the following angle and 2 sides 18

19 (4) isect an angle (a) Given an angle. (d) Do the same as step (c) but placing your pointer at point. Label the intersection D. D (b) reate an arc of any size, such that it intersects both rays of the angle. Label those points and. (e) reate D. D is the angle bisector. D (c) Leaving the compass the same measurement, place your pointer on point and create an arc in the interior of the angle. (f) D is the angle bisector. o o D 1. isect the given angles: 19

20 2. onstruct and angle that is 1.5 the angle: 4. onstruct an angle that is 2.5 the angle. 20

21 (5) onstruct the perpendicular bisector of a line segment (a) Given (b) Place your pointer at, extend your compass so that the distance exceeds half way. reate an arc. (c) Without changing your compass measurement, place your point at and create the same arc. The two arcs will intersect. Label those points and D. D (d) Place your straightedge on the paper and created. (e) Dis the perpendicular bisector of. D M D M 1. onstruct the perpendicular bisectors and D a) b) 21

22 2. onstruct the perpendicular bisectors and D a) c) 3. Given & D. onstruct the perpendicular bisectors and D D 22

23 (6) onstruct a line perpendicular to a given line through a point not on the line. (a) Given a point not on the line. (b) Place your pointer on point, and extend It so that it will intersect with the line in two places. Label the intersections points and. (c) Using the same distance, place your pointer on point and create an arc on the opposite side of point. (d) Do the same things as step (c) but placing your pointer on point. Label the intersection of the two arcs as point D. (e) reate D (f) D is perpendicular to the given line through point. D D D 23

24 1. onstruct a line perpendicular to a given line through a point not on the line. 2. onstruct a line perpendicular to a given line through a point not on the line: 3. onstruct a line perpendicular to a given line through a point not on the line: 24

25 ltitude is a line which passes through a vertex of a triangle and meets the opposite side at right angles. 1. onstruct an altitude to 2. onstruct an altitude to 3. onstruct an altitude to 4. onstruct an altitude to 25

26 5. onstruct an altitude to 6. onstruct an altitude to 7. onstruct an altitude to 8. onstruct an altitude to 26

27 1. Use a compass and a straightedge to construct the following reflections. R m m 2. Use a compass and a straightedge to construct the following reflections. R m m 27

28 3. Use a compass and a straightedge to construct the following reflections. R m m 4. Use a compass and a straightedge to construct the following reflections. R m m 28

29 5. Use a compass and a straightedge to construct the following reflections. R m m 6. Determine the Line of Reflection In trying to find the line of reflection you need to work backwards through the definition of a reflection. onstruct the line of reflection of Δ & Δ How do you know that this is a reflection and not a rotation ' ' ' 29

30 7. Determine the Line of Reflection What about this transformation tells you that it must be a reflection and not something else? onstruct the line of reflection of Δ & Δ. ' ' ' 8. Determine the Line of Reflection What in this diagram gives us a clue about where the line of reflection is? onstruct the line of reflection of Δ & Δ. ' ' ' 30

31 (7) onstruct a line perpendicular to a given segment through a point on the line. (a) Given a point on a line. (b) Place your pointer a point. reate arcs equal distant from on both sides using any distance. Label the intersection points and. (c) Place your pointer on point and extend it past. reate an arc above and below point. (d) Place your pointer on point and using the same distance, create an arc above and below. Label the intersections as points D and E. (e) reate DE. f) DE is perpendicular to the line through. D D D E E E 1. onstruct a line perpendicular to a given segment through a point on the line. 31

32 2. onstruct a line perpendicular to a given segment through a point on the line 3. onstruct a line perpendicular to a given segment through a point on the line 32

33 3. Given. onstruct the perpendicular bisector. 4. Given, construct the angle bisector, ray D. 5. onstruct a line perpendicular to a given segment through a point: 33

34 6. onstruct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction. H G J 7. onstruct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction. a) b) D F E 8. onstruct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction. 34

35 1. Given sides of a rectangle. onstruct the rectangle. (Hint) We need perpendicular lines through and through M. 2. Given the side of a square. onstruct the square. D 35

36 3. 6. Use the diagram to complete the relationship a) = b) a) = b) a) = b) hoose which construction matches the diagram M o o D 6. a) = b) 10. D ' D a) The Midpoint of b) line through c) bisector d) opy a segment a) opy b) bisector c) bisector d) opy a segment a) opy b) bisector c) bisector d) opy a segment a) The Midpoint of b) line through c) bisector d) opy a segment 11. rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and then explain why it MUST be a rhombus. ' ' D 12. If you are told that MN is the perpendicular bisector of where point M is on. Draw the diagram and completely label it with all known relationships. 5. If you are constructing the perpendicular line through point ( is on the line), determine the next step. Step #1 Place compass at point, and create two intersections & on either side of point. Step #2 Place compass pointer at point and extend its measure beyond and make an arc above and below point. Step #

37 (8) onstruct a line parallel to a given line through a point not on the line. (a) Given a point not on the line. (b) Place your pointer at point and measure from to. Now place your pointer at and use that distance to create an arc. Label that intersection D. (c) Using that same distance, place your pointer at point, and create an arc as shown. D D (d) Now place your pointer at, and measure the distance from to. Using that distance, place your pointer at D and create an arc that intersects the one already created. Label that point E. (e) reate E. (f) E is parallel to E E E D D D 1. onstruct a line parallel to a given line through a point not on the line. 37

38 2. onstruct a line parallel to a given line through a point not on the line. 3. onstruct a line parallel to a given line through a point not on the line. 4. onstruct a line parallel to a given line through a point not on the line. 38

39 To inscribe a hexagon in a circle, copy the radius and then copy it around the circumference six times, then connect all points on the circle. To inscribe an equilateral triangle in a circle, copy the radius and then copy it around the circumference six times, then connect every other points on the circle. To inscribe a square in a circle, construct the perpendicular bisector of a diameter then connect the four points on the circle. 1. Inscribe a regular hexagon in a circle by construction. 2. Inscribe an equilateral triangle in a circle by construction. 39

40 3. Inscribe a square in a circle by construction. 4. Inscribe a regular hexagon in a circle by construction. 40

41 5. Inscribe an equilateral triangle in a circle by construction. 6. Inscribe a square in a circle by construction. 41

42 ircumcenter or ircumscribe perpendicular bisectors Incenter or Inscribed angle bisectors Orthocenter altitudes entroid medians 1. ircumscribe a circle about each triangle 42

43 2. Inscribe a circle in each triangle. 3. Locate the orthocenter of each triangle. 43

44 4. Locate the centroid of each triangle. 5. ircumscribe a circle about each triangle 44

45 6. Inscribe a circle in each triangle. 7. Locate the orthocenter of each triangle. 45

46 8. Locate the centroid of each triangle. 9. ircumscribe a circle about each triangle 46

47 10. Inscribe a circle in each triangle. 11. Locate the orthocenter of each triangle. 47

48 12. Locate the centroid of each triangle. 13. ircumscribe a circle about each triangle 48

49 14. Inscribe a circle in each triangle. 15. Locate the orthocenter of each triangle. 49

50 16. Locate the centroid of each triangle. Regents Questions 1. Which illustration shows the correct construction of an angle bisector? 50

51 2. Which diagram shows a construction of a 45 ο angle? 3. onstruct the angle bisector of the given angle. 4. On the diagram below, use a compass and straightedge to construct the bisector of. [Leave all construction marks.] 51

52 5. Using a compass and straightedge, construct the angle bisector of shown below. [Leave all construction marks.] 6. On the diagram below, use a compass and straightedge to construct the bisector of. [Leave all construction marks.] 7. Using only a ruler and compass, construct the bisector of angle in the accompanying diagram. 52

53 8. Using a compass and straightedge, construct the bisector of. [Leave all construction marks.] 9. Using a compass and straightedge, construct an equilateral triangle with as a side. Using this triangle, construct a 30 angle with its vertex at. [Leave all construction marks.] 10. The diagram below shows the construction of the bisector of. Which statement is not true? 1) m EF) = 1 2 m 3) m EF) = m 53

54 2) m DF) = 1 2 m 4) m DF) = m EF 11. straightedge and compass were used to create the construction below. rc EF was drawn from point, and arcs with equal radii were drawn from E and F. Which statement is false? 1) m D = m D 3) 2(m D) = m 2) 1 4) 2(m ) = m D (m ) = m D ased on the construction below, which statement must be true? 1) 1 m D = 2 m D 3) m D = m 2) m D = m D 4) 1 m D = 2 m D 12. student used a compass and a straightedge to construct E in Δ as shown below. 54

55 Which statement must always be true for this construction? 1) E E 3) E E 2) E E 4) E 14. s shown in the diagram below of Δ, a compass is used to find points D and E, equidistant from point. Next, the compass is used to find point F, equidistant from points D and uuu r E. Finally, a straightedge is used to draw F. Then, point G, the intersection of and side of Δ, is labeled. Which statement must be true? 1) 2) uuu r F bisects side uuu r F bisects uuu r 3) F 4) Δ G Δ G 1. Line segment is shown in the diagram below. Which two sets of construction marks, labeled I, II, III, and IV, are part of the construction of the perpendicular bisector of line segment? 1) I and II 2) I and III 3) II and III 4) II and IV 2. One step in a construction uses the endpoints of to create arcs with the same radii. The arcs intersect above and below the segment. What is the relationship of and the line connecting the points of intersection of these arcs? 1) collinear 2) congruent 55

56 3) parallel 4) perpendicular 4. The diagram below shows the construction of the perpendicular bisector of Which statement is not true? 1) = 2) = ½ 3) = 2 4) + = 5. ased on the construction below, which conclusion is not always true? 6. Using a compass and straightedge, construct the perpendicular bisector of. [Leave all construction marks.] 7. Using only a compass and a straightedge, construct the perpendicular bisector of and label it c. [Leave all construction marks.] 56

57 8. Using a compass and straightedge, construct the perpendicular bisector of shown below. Show all construction marks. 9. On the diagram of shown below, use a compass and straightedge to construct the perpendicular bisector of. [Leave all construction marks.] 10. Using a compass and straightedge, construct the perpendicular bisector of side in shown below. [Leave all construction marks.] 57

58 11. Use a compass and straightedge to divide line segment below into four congruent parts. [Leave all construction marks.] 1. The diagram below illustrates the construction of parallel to through point P. Which statement justifies this construction? 2. Which geometric principle is used to justify the construction below? 58

59 3. The diagram below shows the construction of through point P parallel to. 4. The diagram below shows the construction of line m, parallel to line, through point P. 5. The diagram below shows the construction of a line through point P perpendicular to line m. 59

60 6. In the accompanying diagram of a construction, what does represent? 7. Using a compass and straightedge, construct a line that passes through point P and is perpendicular to line m. [Leave all construction marks.] 8. Using a compass and straightedge, construct the line that is perpendicular to and that passes through point P. Show all construction marks. 60

61 9. Using a compass and straightedge, construct a line perpendicular to through point P. [Leave all construction marks.] 10. Using a compass and straightedge, construct a line perpendicular to line through point P. [Leave all construction marks.] 1. Which diagram shows the construction of an equilateral triangle? 1) 2) 3) 4) 61

62 2. Which diagram represents a correct construction of equilateral, given side? 1) 2) 3) 4) 3. On the line segment below, use a compass and straightedge to construct equilateral triangle. [Leave all construction marks.] 62

63 4. Using a compass and straightedge, and below, construct an equilateral triangle with all sides congruent to. [Leave all construction marks.] 5. Using a compass and straightedge, on the diagram below of, construct an equilateral triangle with as one side. [Leave all construction marks.] 6. On the ray drawn below, using a compass and straightedge, construct an equilateral triangle with a vertex at R. The length of a side of the triangle must be equal to a length of the diagonal of rectangle D. 63

64 64

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