CONSTRUCTIONS
Table of Contents Constructions Day 1...... Pages 1-5 HW: Page 6 Constructions Day 2.... Pages 7-14 HW: Page 15 Constructions Day 3.... Pages 16-21 HW: Pages 22-24 Constructions Day 4.... Pages 25-29 HW: Pages 29-30 Construction Project.... Pages 31-32 Due Date for project: For this unit you will have to know how to: 1. Construct an equilateral triangle, using a straightedge and compass, and justify the construction 2. Construct a bisector of a given angle, using a straightedge and compass, and justify the construction 3. Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction 4. Construct lines parallel to a given line through a given point, using a straightedge and compass, and justify the construction 5. Construct lines perpendicular to a given line through a given point, using a straightedge and compass, and justify the construction
Constructions Day 1 1
2
3
4
Challenge SUMMRY 1. Exit Ticket 2. 5
Homework Day 1 1. 2. 3. 4. 6
Constructions Day 2 Warm - Up 7
8
On the accompanying diagram of BC, use a compass and a straightedge to construct an altitude from to BC. 9
10
11
Challenge 12
SUMMRY 13
Exit Ticket 1. 2. 14
Homework Day 2 1. 2. 3. 4. 5. 6. 15
Constructions Day 3 CONCEPT 1 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 1. The construction of an inscribed equilateral. () Given Circle (B) Create a diameter BC (C) Create a circle at C with radius C. Label the two intersections D and E. B B E C D C 16
(D) Create BD, BE & ED (E) The inscribed Equilateral B B E E D C D So why does this work? B Reason #1 In Step (C) we form two equilateral triangles, DC & EC because of the three congruent radii. Equilateral triangles have three 60 angles, so m DC = m EC = 60 and m DE = 120 which is the central angle. This has cut the circle exactly into thirds (360 / 120 = 3). D 120 60 120 60 60 60 60 60 C E mde meb mbd, thus chords DE EB BD. Reason #2 In Step (C) we formed a special right triangle of 30 60 90 because E is a right angle (inscribed on a diameter) and BC = 2EC (d = 2r), which only happens in the special right triangle of 30-60-90. B r E This makes m EBC = 30, m DBC = 30 and m EBD = 60. That makes EBD an isosceles triangle with a vertex angle of 60, and base angles of 60. D r r C 17
You Try it! Construct an inscribed equilateral. 2. The construction of an inscribed square. () Given Circle (B) Create a diameter BC (C) Construct a perpendicular line to BC through. B B D C E C (D) Create BD, DC, CE & EB B (E) The inscribed Square B D D E C E C 18
So why does this work? Reason #1 Square has diagonals that are perpendicular and congruent. The perpendicular diameters determine the square. Reason #2 The perpendicular bisectors form a central angle of 90 which divide the circle into 4 congruent parts, thus forming the square. You Try it! Construct an inscribed square. 3. The construction of an inscribed hexagon. () Given Circle (B) Create a diameter BC (C) Create a circle at C with radius C. Label the two intersections D and E. B B E C D C 19
(D) Create a circle at B with radius B. Label the two intersections F and G. (E) Create CD, DF, FB, BG, GE & EC (F) The inscribed Hexagon B G B G E E B G F F E D C D C F D C So why does this work? Reason #1 Step (D) divided the circle into 6 congruent arcs, thus six congruent chords. Reason #2 Step (D) created six equilateral triangles, FB. BG, GE, EC, CD, and DF) dividing the circle into six congruent parts. You Try it! Construct an inscribed hexagon. 20
Exit Ticket 21
Day 3 HW 1. Determine whether the relationships is INSCRIBED or CIRCUMSCRIBED. a) The triangle is. b) The hexagon is c) The circle is d) The hexagon is e) The circle is f) The triangle is 2. Jeff uses his compass to make a cool design. He just keeps creating congruent circles over and over a) Find a regular hexagon (shade it in) b) Find a different regular hexagon (shade it in) c) Find an equilateral triangle (shade it in) d) Find a different equilateral triangle (shade it in) 22
3. The inscribed equilateral triangle has a central angle of 120 because 360 / 3 = 120, an inscribed square has a central angle of 90 because 360 / 4 = 90. The central angle of a decagon is 36 because 360 / 10 = 36. Use this information and a compass to create an inscribed decagon. 36 4. Construct the requested inscribed polygons. a) Construct an equilateral triangle inscribed in the provided circle using your compass and straightedge. b) Construct a square inscribed in the provided circle using your compass and straightedge. 23
5. Construct the requested inscribed polygons. a) Construct a regular hexagon inscribed in the provided circle using your compass and straightedge. b) Construct a regular octagon inscribed in the provided circle using your compass and straightedge. Hint: The central angle is 45, half of the square s central angle of 90. 24
Constructions Day 4 Warm Up: 25
26
27
28
Day 4 - HW 29
2. 3. Inscribed Polygons a) Inscribed Square b) Inscribed Equilateral Triangle c) Inscribed Hexagon 30
Construction Project In this project you will use your knowledge of constructions to create a booklet, poster, or study guide that someone could use to learn this skill. You must demonstrate your skill at performing the 10 constructions we learned in class as well as explain, step-by-step, how to do each construction. 5 points 30 points The constructions you are responsible for are: 1) Construct the perpendicular bisector of a line segment. 2) Construct the bisector of an angle 3) Construct an equilateral triangle. 4) Construct an angle congruent to a given angle. 5) Construct a line parallel to a given line through a given point. 6) Construct a line perpendicular to a given line through a given point that is not on the given line. 7) Construct a line perpendicular to a given line through a given point that is on the given line. 8) Construct all three Inscribed Polygons. 31
For each construction (There are seven) you must complete 2 tasks: Task 1: Do the construction in its entirety: Task 2: Create a step-by-step explanation for each construction in your own words: 32