Unit 2: Constructions

Transcription

1 Name: Geometry Period Unit 2: Constructions In this unit you must bring the following materials with you to class every day: COMPASS Straightedge (this is a ruler) Pencil This Booklet A device Headphones Please note: You may have random material checks in class Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class everyday. All homework for this unit is in this booklet. Answer keys will be posted as usual for each daily lesson. 1

2 Constructing an Inscribed Hexagon Today s Goal: What are constructions? How do we use the geometric compass? What is some new vocabulary related to constructions? How do I construct an inscribed hexagon? How do I construct an inscribed equilateral triangle? What are geometric constructions? 2-1 Notes Tips for using our compass! What makes a good construction? Visualize what the end result should look like before you begin! Leave all construction marks! Full circles or arcs are fine! You must PRACTICE PRACTICE PRACTICE! 2

3 Before we construct, let s kick start your knowledge! *Did you know? The side length of a hexagon is equal to its circumradius - the distance from the to a. (This is also the.) Before we Construct! FACT CHECK What are the defining characteristics of a regular hexagon? Which of the following is an example of an inscribed hexagon? a) b) c) What does it mean for a hexagon to be inscribed? 3

4 Let s Construct! Construct a regular hexagon in the space below: Keep it small it has to fit in this box! Take it to the next level! Sketch an equilateral triangle inscribed in a circle Fact check: What are key features of an equilateral triangle? Using our construction of a hexagon how can form an equilateral triangle inscribed in a circle? Make construction marks (just like a hexagon) then 4

5 Try it!! Construct a hexagon inscribed in a circle Explain each step you took to complete your construction 1. Set compass to the same measure of the radius 2. Plot any point on the circle to start 3. Put point of compass on the point and draw an arc the length of the radius 4.Place compass where you just sketch an arc and draw another arc. Repeat this step until you have 6 arcs in total around the circle 5. Use a straightedge to construct each side of the hexagon 6. All done! Construct an equilateral triangle ABC inscribed in a circle Explain each step you took to complete your construction 1. Draw circle with a center at point P 2. Set compass to the same measure of the radius Put point of compass on the point and draw an arc the length of the radius

6 2-1 Practice 1. What type of hexagon are we constructing in this lesson? What does it mean for a geometric figure to be regular? (Use your device to look up the definition for regular, with respect to geometric figures.) 2. Error analysis! Analyze the student work to the right for parts a-c: a. Identify what the student did well. b. Identify where the student made a mistake. Be specific when describing what the mistake was. c. With a different colored pen/pencil and the given arc marks, correctly complete the construction of an inscribed hexagon. 3. How can you expand your construction of a regular hexagon to construct a regular dodecagon with twelve congruent sides? Try it below! STOP here! Check the key in a different color! 6

7 2-1 Homework Complete each of the following problems. Check your work on the website when you re done. 1. a) How can you construct an equilateral triangle using the construction of an inscribed hexagon? b) Construct an inscribed equilateral triangle inside circle p. 2) Watch the assigned video and try your constructions on this page. Mastery of the content of this video is essential for our next lesson in class. Failure to watch the video will result in confusion and your inability to interact with your peers throughout the lesson. This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery of the concept. Video on Edpuzzle! Sign in through Google!! a) What are we constructing in the video? b) What is special about this geometric figure? Try it again! Jot it down! What are the steps you took in this construction? (Use numbered steps) 7

8 Apply It! 3) Mr. Gino has three cats. He has heard that cats in a room position themselves at equal distances from one another and wants to test that theory. Mr. Gino notices that Simon, his tabby cat, is in the center of his bed (at S), while Snowball, his Siamese, is lying on his desk chair (at J). If the theory is true, where will he find Checkers, his tuxedo cat? Use the scale drawing of Mr. Gino s room shown below, together with (only) a compass and straightedge. Place an M where Isosceles will be if the theory is true. 4) Construct equilateral triangle QRS with side length MH and a vertex at R: R 8

9 2-2 Notes Perpendicular Bisector and Inscribed Square Today s Goal: 1) How do I construct a perpendicular bisector through a given segment? 2) How do I construct a square inscribed in a circle? Conceptual Understanding 1) Harrison has two city parks, P1 and P2. The city council would like to add a Starbucks in town (YAY!), and would like the Starbucks to be an equal distance to both parks. Identify a few possible locations for the Starbucks, and label them as S 1, S 2, S 3, etc. on the map. Let s Discuss: With your elbow partner, what geometric details do you see in your construction? 9

10 Part I: Perpendicular Bisector: VIDEO TIME! Let s watch our next construction before moving forward! a) What are we constructing in the video? b) What is special about this geometric figure? 1. Try it here! 2. Extend your thinking: Locate the midpoint on one of the segments above and label it M. Try another! Construct the perpendicular bisector of side AC in the triangle below. 10

11 Part II: Inscribed Square Construction Before we construct! FACT CHECK Squares Imagine it! In the box below, SKETCH what a square inscribed in a circle would look like: Now, sketch in the diagonals of that square. What are the diagonals the same as in the circle? Given the following circles, try to construct an inscribed square. Extra Circles provided for scrap work! Take 1 Perfection Take 2 List the steps you took to construct an inscribed square: 1) Start by drawing in the with a straight edge 2) Construct the of the segment from step 1 3) Draw vertices where your two segment intercept the circle, and connect 4 sides using your straightedge. 11

12 Part III: Mixed Construction Practice 1) ABC is shown below. Is it an equilateral triangle? Justify your response (NO MEASURNG WITH RULERS!) 2) Construct equilateral triangle QRS with side length MT and a vertex at R: R 12

13 3) a. On the coordinate plane, plot the points M(1,2) and N(-1,4) b. Construct the perpendicular bisector between those two points c. Algebra review: Write the equation of the line you just constructed.. 4) Using your knowledge of what you know how to construct at this point, explain how you would (include sketches) Construct a Midpoint Construct a Right Angle Construct a 60 angle 5) Construct 3 different polygons inscribed in a circle: 13

14 6) In a square diagonals are of each other. 7) When a polygon is in a circle, all of its vertices are ON the circle. 8) Use a compass and straightedge to divide line segment AB below into four congruent parts. (Leave all construction marks) 9) 14

15 2-3 Notes Circumscribe a Circle Around a Triangle (circle around a triangle) Today s Goal: What is and how do we locate the circumcenter of a triangle? How do I construct a circle circumscribed around a triangle? Let s get to work! Do now on your own! In the following triangle, construct the perpendicular bisectors of sides AB and BC Hint: Turn your paper to make it easier. What type of point did you construct on side AB? What type of point did you construct on side BC? Locate the point that the two perpendicular bisectors intersect at and label it R Use the ruler provided to measure: With an elbow partner! a) The distance between vertex A and point R. b) The distance between vertex B and point R c) The distance between vertex C and point R What conclusion can be made about these distances? 15

16 Together! When two lines or more intersect at the same point, this point is called a point of. You ve just constructed a point of concurrence called the. FACT CHECK Circumcenter- Using technology! Formed when two or more Intersect. Equidistant from of the triangle. Can be found, or of the triangle. Complex Constructions that involve the Circumcenter Circle circumscribed around a triangle To circumscribe means to. Based on the definition above, if you are circumscribing a triangle, what shape do you expect to be constructing? Before we construct! FACT CHECK: CIRCLES Center: Radius: A segment that extends from of the circle to the edge of the circle The lengths of all radii in a circle are 16

17 Analyze the Following Step-by-Step Construction With your elbow partners, examine the steps used when constructing a circle circumscribed around a triangle, then answer the following questions: Step1 Step 2 Step 3 Step 4 Step 5 1. What simple construction happened in steps 1 and 2? 2. Based on your answer from question 1, what type of point is point O in the construction step 3? Explain! 3. In step 5, if we extended a segment from O to C, what type of segment is OC? Quick Summary! A circumcenter is formed by constructing the or more sides of a triangle. of two A circumcenter helps us construct. The circumcenter is from the vertices of the triangle. 17

18 Practice 1. Watch the video of a circle circumscribing a triangle on EdPuzzle. 2. Try it! Either try with the video, or on your own the following: Construct a circle that is circumscribed around triangle ABC. ( or the triangle inscribed in the circle) Remember! A circumcenter can be inside, on, or outside the triangle! 3. What is the circumcenter of the triangle whose vertices are A(-7,0), B(-3,8), AND C(-3,0)? 18

19 4. A martial arts expert is standing at point E in a triangular ring with three of his equally talented students that are standing at A, B and C. The worst place he could stand is where the three students could deliver a chop or leg kick to the expert at the same time. Which point of concurrency would represent this worst place he could stand? 5. Construct a circle circumscribed around triangle MNO. 19 STOP here! Check the key in a different color!

20 2-3 Homework Complete each of the following problems. Check your work on the website when you re done. 1. Answer the following based on the class notes from 2-3 and new vocabulary: a) What is the name of the point of concurrence that is equidistant from the three vertices of the triangle. b) What type of segment needs to be constructed twice to locate this point? c) Where can this point be located with respect to a triangle? 2. A. What type of construction is shown below? b. Solve for z. 3. A. What part of the circle are segments AB and BG? b. What is the relationship between AB and BG? c. Hence, if AB = 5y - 6 and BG = 24, solve for y. 20

21 Watch the assigned video and try your constructions on this page. Mastery of the content of this video is essential for our next lesson in class. Failure to watch the video will result in confusion and your inability to interact with your peers throughout the lesson. This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery of the concept. Video on Edpuzzle! Sign in through Google!! a) What are we constructing in the video? Try it! Construct lines perpendicular to the given line through point P in both examples. P P Jot it down! What are the steps you took in this construction? (Use numbered steps) 21

22 2-4 Notes Constructing Altitudes and Squares (using side length) Today s Goal: How do I construct altitude and a square (given a side length)? Explain each step you took to complete your construction 1. Place needle point on point P and swing the compass through segment AB so that it crosses the segment twice. Construct the line perpendicular to AB through point P: 2. Label the two points of intersection, Q and R. 3. Construct a perpendicular bisector of segment QR. 4. The constructed line should pass through point P; This is the line perpendicular to AB thorough point P Push Your Skills Construct any size right triangle BEF: How do the constructions on this page compare to the construction from your check in/video last night? 22

23 Before we construct! FACT CHECK: What is an altitude of a triangle? In Words Sketch a line that extends from of a triangle and is to the side. Considering your sketch, what will the construction of an altitude require? Let s try this together: Construct an altitude in Triangle MNO label it MP Steps: 1. Place needle point on point M and swing the compass through segment NO so that it crosses the segment twice. 2. Label the two points of intersection, A and B. 3. Construct a perpendicular bisector of segment AB. 4. The constructed line should pass through point M; this is the line perpendicular to BC thorough point x. 5. This is the altitude of the 23

24 List the properties of a square. With your shoulder partner, discuss how these properties can be used in the construction of a square. Properties of a Square Diagonals are perpendicular bisectors Adjacent sides are perpendicular Construction(s) that would lead to this property -Perpendicular Bisector Construction (Use when inscribed in a circle) Visualize All sides are congruent Construct a square given a side length 1) Watch me 2) Try on your own 3) Follow steps provided Constructing a Square 1. Extend line segment AB using a straightedge on side A. 2. Construct the perpendicular line at point A (just like yesterday s class) 3. Measure the distance from A to B with your compass. 4. Needlepoint on A; swing arc above the line segment to mark distance of AB. Mark the intersection of this arc with the perpendicular line. Label D. (extend perpendicular line with straightedge if necessary) 5. Needlepoint on B; swing arc above the line segment to mark distance of AB. 6. Needlepoint point C; swing compass to cross at the arc made in step 5. Label the intersection C. 7. Draw segment CD and BC. Done! 24

25 Let s Construct! Using only your compass and a straight edge, construct a square with side length XY: X Y You Try! Using only your compass and a straight edge, construct a square with side length AB: 25

26 2-4 Homework Complete each of the following problems. Check your work on the website when you re done. 1) Using a compass and a straightedge, construct the line that is perpendicular to the given line and passes through point p. 2) ON the line below, construct square MATH whose side length is equal to AB 3) The diagram below shows the construction of a line through point P perpendicular to line m. Which statement is demonstrated by this construction? a) If a line is parallel to a line that is perpendicular to a third line, then the line is also perpendicular to the third line. b) The set of points equidistant from the endpoints of a line segment is the perpendicular bisector of the segment. c) Two lines are perpendicular if they are equidistant from a given point. d) Two lines are perpendicular if they intersect to form a vertical line. 26

27 4) a) In the accompanying diagram of a construction, what type of special segment INSIDE a triangle, does represent? b) What is the relationship between PC and AB? 5) Look at the construction of a square below. Identify two parts of the square and describe the specific constructions that you might be able to use to create a square. (For example, side BC can be created with a straightedge. [This is just an example. Yours should be more involved]) D C 6) In triangle PQR, using a compass, construct an altitude from vertex P to side QR. 27

28 2-5 Notes UNIT 2: CONSTRUCTIONS PROGRESS REPORT You will work on a blend of concepts from this unit so far! Stuck? Be resourceful, check your notes, look at videos from class and from EdPuzzle, and use your peers! Before we start let s check out the square one more time Together: Using only your compass and a straight edge, construct a square with side length AB: Complete this ENTIRE SECTION for HOMEWORK! (STOP at 2-6) 28

29 Construct an Equilateral Triangle whose side length is the length of AB Construct an Equilateral triangle whose sides are the same length as AC on the line below. Construct a line Perpendicular to the given line through point p Construct an Altitude in the following triangle through vertex B 29

30 Construct a square inscribed in circle Construct the midpoint of side AC, label it M Construct an inscribed Hexagon Construct an Inscribed Equilateral Triangle 30

31 Construct the circumcenter in the following triangle, Label it Q Circumscribe a circle around the following triangle. Construct a line through P perpendicular to OP *Note, your line should be outside of the circle not in!! Construct the altitude of the following trapezoid through point B 31

32 Watch the assigned video and try your constructions on this page. Mastery of the content of this video is essential for our next lesson in class. Failure to watch the video will result in confusion and your inability to interact with your peers throughout the lesson. This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery of the concept. Video on EdPuzzle!! a) What are we constructing in the video? Video Practice: b) Try it again! Try it again! 32

33 2-6 Notes Constructing an Angle Bisector Today s Goal: How do I construct an angle bisector? How can we use our knowledge of angle bisectors to construct extension constructions? Let s revisit the concept we learned last night!! Angle Bisector Steps 1. Using any setting, place compass point at O and swing an arc through both sides of the angle. Label the intersection with P and Q. Bisect DOB. Show all arc marks 2. Using any setting, place compass point at P and swing an arc. Using same compass setting do the same from point Q. Label intersection point R. 3. Draw OR. 1. What is an angle bisector? Group Talk: In your groups discuss the following questions be prepared to share out! 2. How would you construct a 60 0 angle? How would you construct a 30 0 angle? Sketch your thinking here! 3. How would you construct a 45 0 angle? Be specific (Number your steps) Sketch your thinking here! 33

34 Using our answers from our share out, as a team, complete the following examples! Example 1) Example 2) Using a compass and a straightedge, construct an Using a compass and a straightedge, construct a 45 equilateral triangle with side AB as a side. angle with its vertex at the midpoint of segment AB. Using this triangle, construct a 30 angle with its vertex (Leave all construction marks.) LABEL THIS ANGLE! at A. (Leave all construction marks.) LABEL THIS ANGLE! Still have time? Try to create a 30 angle a different way! (Sketch in your own starting points/segments first) Question: What do math teachers think of owl of their students? 34

35 Challenge yourself! 1. Construct a 60 degree angle in 3 different ways! (Draw your own line segments/points to start) 2. a) Inscribe a square in the circle d b) Construct a perpendicular bisector of any side of your square c) Mark the intersection point of the perpendicular bisector and that side of the square with a Z d) With a straightedge, connect point Z with the opposite two vertices. e) Describe what you constructed Complete ALL problems on this page. Make sure you CHECK YOUR HW!!! 35

36 2-6 Homework 1. Construct a circle circumscribed around the triangle shown below. 2. The measure of one interior angle of a regular pentagon is 108 degrees. Construct a 27 degree angle. 36

37 3) 4) a. What types of segments are constructed inside of the triangle below? Use the markings to help you. b. If <BAP = 2x and <CAP = 46, solve for x. c. What is the measure of angle A? 37