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
SUMMARY
Exit Ticket
1.
2.
5
Homework – Day 1
1.
3.
2.
4.
6
Constructions – Day 2
Warm - Up
7
8
On the accompanying diagram of ∆ABC, use a compass and a straightedge to
construct an altitude from A to ̅̅̅̅
𝐵𝐶 .
9
10
11
Challenge
12
SUMMARY
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.
(A) Given Circle A
(B) Create a diameter BC
(C) Create a circle at C with radius AC
. Label the two intersections D and E.
B
B
A
A
E
A
D
C
C
16
(E) The inscribed Equilateral
(D) Create BD , BE & ED
B
B
A
E
A
D
E
C
D
So why does this work?
B
Reason #1 – In Step (C) we form two equilateral triangles,
DAC & EAC because of the three congruent radii.
A
E
120°
120°
Equilateral triangles have three 60° angles, so mDAC =
mEAC = 60° and mDAE = 120° which is the central angle.
This has cut the circle exactly into thirds (360° / 120° = 3).
̂ ≅ 𝑚𝐸𝐵
̂ ≅
𝑚𝐷𝐸
̂,
𝑚𝐵𝐷
60°
60°
60°
60°
60°
60°
D
C
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.
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.
B
r
A
E
r
r
D
C
17
You Try it! Construct an inscribed equilateral.
A
2. The construction of an inscribed square.
(A) Given Circle A
(C) Construct a perpendicular line to
(B) Create a diameter BC
BC through A.
B
B
D
A
A
A
E
C
(D) Create BD , DC , CE & EB
C
(E) The inscribed Square
B
B
D
D
A
A
E
E
C
C
18
So why does this work?
Reason #1 – A 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.
A
3. The construction of an inscribed hexagon.
(A) Given Circle A
(B) Create a diameter BC
(C) Create a circle at C with radius AC
. Label the two intersections D and E.
B
B
A
A
E
A
D
C
C
19
(D) Create a circle at B with radius BA
. Label the two intersections F and G.
B
(E) Create CD , DF , FB , BG , GE
(F) The inscribed Hexagon
& EC
B
G
G
B
A
A
E
F
G
E
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, FAB. BAG, GAE, EAC, CAD, and
DAF) dividing the circle into six congruent parts.
You Try it! Construct an inscribed hexagon.
A
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