Geometry Constructions

Transcription

1 age 1 Geoetry Constructions Nae: eriod:

2 age 2 Geoetric Constructions Construct a segent congruent to a given segent Given: B Construct a segent congruent to B 1. Use a straightedge to draw a segent longer than the given segent. Label a point R at one endpoint of the new segent. 2. lace the copass tip at point of the given segent. djust your copass width to equal the length of B. 3. Using this sae copass setting, place the copass tip at point R and draw an arc. Label the intersection point S. 4. Erase the excess segent. 5. B RS B Construct a segent congruent to B B B B B

3 age 3 Construct an angle congruent to a given angle Given: Construct an angle congruent to. 3. Draw a ray. Label the endpoint D. 4. lace the copass tip at the vertex of. Draw an arc across both sides of the given angle. Label the points of intersection with the rays B and C. 3. Using this sae copass setting, place the copass tip at point D (the new ray) and draw a long arc across the ray. Label the intersection point E. 5. Set the copass so that it is the width of BC. 5. Using this sae copass setting, place the copass tip at point E and draw an arc, intersecting the arc fro step 3. D Label the intersection F. 6. DrawDF. EDF BC Construct an angle congruent to. 1. Your construction here: 2.

4 age

5 age 5 erpendicular Bisector Given: B Construct the perpendicular bisector of B. 1. Choose a copass opening greater than 1/2 of B and less than the length of B. lace copass tip at. Draw two arcs - above and below B. 2. Using the sae copass opening, place copass tip at point B. Draw two arcs above and below B. 3. Draw the line connecting the intersections of the two arcs. This is the perpendicular bisector ofb. (Can also be used to find the idpoint of B.) B Construct the perpendicular bisector of each of the following line segents C B D E B F

6 age 6 RCTICE Construct the segent or angle that is congruent to the given segent or angle

7 age 7 ngle Bisector Given:. Construct the angle bisector of. 1. lace the copass tip at point. Draw an arc that intersects both rays of the angle. Label the points of intersection B and C. 2. lace the copass tip at point B and draw an arc in the interior of. 3. Using this sae copass setting, place the copass tip at point C and draw an arc that intersects the arc you drew in #2. Label the point of intersection Q. 6. Use a straightedge to draw Q. * This is the angle bisector of. 7. BQ QC Construct the angle bisectors for each of the following angles

8 age 8 arallel Lines Given: oint X and line Construct a line parallel to line containing point X. 1. lace point anywhere on line. Draw line X. 2. lace copass tip at point and draw an arc that intersects line and line X Label the intersection point Q with line X. Label the intersection point B with line. 3. Using the sae copass opening, place copass tip at point X and draw an arc of the sae size. Label the intersection point T (on line X ). 4. lace the copass tip at point B. djust the opening so that the pencil tip is where the arc intersects line X at point Q and draw a little arc. 8. Using the sae copass opening, place the copass tip at point T. Draw an arc fro this point that intersects the arc you drew in #3. Label the point of intersection R. 9. Draw line XR. XR X Construct a line parallel to line at point X in the following probles X X

9 age 9 RCTICE Construct a line parallel to the line given through point Construct the segent or angle bisector for each segent or angle

10 age 10 erpendiculars, Given a oint ON the Line Given: Line which contains oint Construct a perpendicular to line through point. 1. lace copass tip at point. Using any copass opening less than the length of, draw two arcs intersecting line on both sides of. Label these points C and D. 2. lace copass tip at point C. djust the copass so that it is greater than ½CD and draw an arc above. 3. Using the sae copass opening, place copass point at point D. Draw an arc above intersecting your first arc. 4. Label the intersection point X. 5. Use a straightedge to draw line X. This is perpendicular to through point. Construct perpendiculars to line through point

11 age 11 erpendiculars, Given a oint NOT on the Line Given: Line and oint not on the line. Construct a perpendicular fro point to line. 1. lace copass tip at point. Using an arbitrary radius, draw arcs Intersecting line at two points. Label these points and B. 2. Using a copass opening greater than ½ B, place copass point at point. Draw an arc below line. 3. Using the sae copass opening, place copass tip at point B and draw an arc fro point B below line, intersecting arc fro step Label the intersection point X. 5. Draw line X. This is perpendicular to through point. Construct perpendicular lines fro point to line

12 age 12 RCTICE Construct a line perpendicular to the line given through the point

13 age 13 Constructions Review Date eriod Construct the following: 1. line parallel to line through point X X 2. The perpendicular bisector of B B 3. The angle bisector of BC B C

14 age The arcs for a copass and straightedge construction are shown below. What construction is apparently being ade? ) Two lines parallel to MN B) Two congruent angles C) segent congruent to MN D) The perpendicular bisector of MN 5. One piece of pie is left for two boys to share. Where should the pie be cut to ensure each gets an equal piece? ) CZ B) CY C) CX D) CW 6. Eric constructed BD as shown. Which of the following stateents ust be true? ) B BC B) BD 2B C) BD CBD D) CBD 2( BC ) 7. Which point is on the line to l and passing through Z? ) U B) V C) W D) X

15 age Which of the following constructions is illustrated? ) The angle congruent to a given angle B) The bisector of a given angle C) The bisector of a given segent D) The perpendicular bisector of a given segent 9. Which line segent is apparently congruent to B? ) D B) C C) E D) F 10. Which segent is congruent to B? ) CK B) CL C) CM D) CN 11. Which point apparently lies on the perpendicular to l fro? ) X B) Y C) Z D) W

16 age Fill in the blanks: a) The perpendicular bisectors eet at the. b) The altitudes eet at the. c) The edians eet at the. d) The angle bisectors eet at the. 13. In each figure below, tell what point of concurrency is illustrated and identify the construction that fors that point. oint: oint: oint: oint: Fored by Fored by Fored by Fored by 14. Given the following pictures and arkings, identify if the dotted line is (a) an angle bisector, (b) a perpendicular bisector, (c) an altitude, or (d) a edian. List all that apply. a. b. a. b. c. c. d. e. d. e. f. f. g. h. g. h. i. i.

17 age 17 Construct the incenter for each triangle.

18 age 18 Construct the circucenter for each triangle.

19 age 19 Merry-go-round 1. The Sith Construction Copany has been hired to install a new water fountain at Winstonian ark. They would like to find the best location for the fountain so that the walking distance fro each of the three ain pieces of playground equipent is the sae. Locate the point and explain how you deterined this. Swing See-Saw 2. The first-aid center of Starved Rock needs to be at a point that is equidistant fro three bike paths that intersect to for a triangle. Locate this point so that in an eergency edical personnel will be able to get any one of the paths by the shortest route possible. Which point of concurrency is it? 3. aula Deene wishes to center a butcher-block table at a location equidistant fro the refrigerator, stove, and sink. Which point of concurrency does aula need to locate? 4. Which point is the circucenter of the triangle? (Be sure to show all work!) B oint Z Y X W C 5. Identify the incenter of the triangle. oint

20 age 20 Refresher: Concurrent Lines of a Triangle Centroid The edians of a triangle are concurrent and intersect each other in a ratio of 2:1. Circucenter erpendicular bisectors of sides of a triangle are concurrent at a point equidistant fro the vertices. Incenter The bisectors of the angles of a triangle eet at a point that is equally distant fro the sides of the triangle. Orthocenter Ortho eans Right The point where the lines containing the altitudes are concurrent is called the orthocenter of a triangle.