Constructions
Constructions
Curriculum Ready
www.mathletics.com
Constructions are all about creating precise, accurate mathematical diagrams.
Compass
Pencil
Rule/straight edge
Equip yourself with these tools and you will soon be creating cool constructions like this:
Center of an Incircle: A circle whose circumference just touches each side of the triangle.
is Give th
Q
a go!
D raw (or construct) an angle below that is exactly 45o at the end point A on the ray AB below using only a compass and a straight edge!
A
B
Work through the book for a great way to do this
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Constructions
Construction Terms To construct geometric shapes or properties, these terms are important and used often. Term
Description
Picture
Line Segment
A straight line with a definite start and end point
A
Ray
A straight line with a definite start and no end point
A
Line
A straight line which continues indefinitely in either direction
Intersection point or Point of intersection
The point where lines (curved or straight) cross each other
Perpendicular lines
Two straight lines that cross each other at exactly 90o
Perpendicular line bisector
A line that forms a 90o angle with an interval and cuts it in half A line that divides an angle in half
Arc
Part of a circle drawn using a compass
Arcs of equal radii (radii is plural for radius)
Two arcs drawn without changing the width of the compass
Point on a line
A particular location on a line, usually labeled with a letter.
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B
A
B
Where lines cross
Angle bisector
An external point
2
A point that is not located on the line or object drawn Constructions Mathletics Passport
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Small box means 90o
A
0
B
Means same length, AO = OB
A A
How does it work?
Constructions
Line construction: Perpendicular bisector The perpendicular bisector is a line that passes through the midpoint of an interval, at a right-angle (90o) to it. A B
1 Set up
2
• D raw two arcs, one above and one below the line segment
• Start with line segment AB A
B
• S et the compass to more than half-way along the line segment
A
A
B
B
3
4 • Rule a line joining the intersections of the arcs
• R epeat step 2 from the other end. Each new arc must cross the first pair
• M ark in the right-angle and call it M (for midpoint)
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For all constructions, these thin, light construct lines must be left on the diagram. It is the ‘working out’ for these types of questions.
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Constructions
Line construction: Perpendicular bisector
A
M
B
..../...
R
1
../20...
BISECTO
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RPENDICUL PE
Draw the perpendicular bisector for the line segments or sides labeled AB below. Show all construct lines used.
RUCTION: ST
INE CO *L N
Your Turn
AR
How does it work?
How does it work?
Constructions
Line construction: Perpendicular line at a point on the line This is a line that is perpendicular to a line, passing through a specific point marked on it. A
P
1 Set up
2
• D raw two arcs across the line AB on either side of point P
• C onstruct a line perpendicular to AB, through point P A
B
B
P
• S et the compass to a radius that would cross the line AB either side of the point P (when the point of the compass is at P)
A
B
P
A
B
P
3
4
• M ake the radius of the compass larger than before
• R ule a line XY joining the intersection of the arcs and the point P
• D raw an arc from each of the intersection points from step 2, making sure they cross each other
• Mark in the right-angle
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Constructions
Line construction: Perpendicular line at a point on the line
T
AR LINE A UL
PE PER NDIC
Draw a perpendicular line XY that cross the line segment/side AB below through the point P. Show all construct lines used.
*
B
THE LINE
2
B
ON
P
A
..../.
A POINT
ON CTI U R T ONS C E LIN ..../20...
1
What special geometrical line have you constructed for this hexagon?
C
P D
A
F
E What is special about the perpendicular to the line AB and the point P for this construction?
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Constructions
Line construction: Perpendicular line through a point external to the line This is a line that is perpendicular to a given line that passes through a specific point external to the line. P B A
1 Set up
2
• Draw two separate arcs across the line AB
• C onstruct a line perpendicular to AB through the external point P P
P
B
A
• M ake the radius of the compass long enough to cross the line AB when the compass is at P
A
B
P A
B
3
4
• Keep the radius of the compass the same • D raw an arc from each of the intersection points from step 2, on the other side opposite point P
• R ule a line joining the intersection of the arcs and the point P • M ark a right-angle where the line crosses the line AB
• Make sure they cross each other 3
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Constructions
ine construction: Perpendicular line through a point external L to the line Draw a perpendicular line XY that cross the line segment/side AB below through the external point P. Show all construct lines used. 1
P
A
B
2
B
P
A
E THRO LIN UG
LI NE
E
O THE L L T IN NA
B
A
P
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/20...
POINT EXTE R
CO NS TR UC T ..../..... ION
A
AR
ERPENDICU *P L
For this construction, one arc will need to pass through the vertex A.
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How does it work?
Constructions
Line construction: Parallel line through a point external to the line This is a line that is parallel to a given line that passes through a specific point external to the line. P A
B
1 Set up
2
• C onstruct a line perpendicular to AB through the external point P P
A
• S et the radius of the compass to less than the distance PQ • D raw a single arc through PQ and through the line AB • D raw another similar arc with the compass at point P
B
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• Rule a line through P and the line AB at an angle 72
62 52 42 32 22 12 02 91 81
P
P
71 61 51 41 31 21 11 01 9 8 7 6 5 4 3 2 1 0
A
Q
A
B
3
Q
B
4
• S et the radius of the compass to equal the distance between the two points of intersection of the first arc
• M ove the compass to point R and draw an arc across it to find point S • Rule a line joining points P and S • PS||AB
R
R
P 82
A
Q
B
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Constructions
Line construction: Parallel line through a point external to the line Construct a line segment XY parallel to AB, passing through the external point P. Show all construct lines used.
LEL LI RAL NE PA
E LIN
P
ION UCT R T S CON
20... / . . . . ..../.
THE LINE *
A 2
XTERNA T E L IN
ROUGH A P TH O
TO
1
B
Use either line to construct a line parallel to JK and LM, that passes through P. J
K
P
L 3
M
Create a parallelogram by constructing lines parallel to the lines AB and CD, both passing through P.
A P
C
D B
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Constructions
Applications of line constructions 1 Follow these steps to find the center of the circle below.
(i) Construct a line perpendicular to the interval AD, passing through the circumference point B. Label the other point where this new perpendicular line crosses the circle circumference E. (ii) Construct another line perpendicular to the interval AD, passing through the circumference point C. Label the other point where this new perpendicular line crosses the circle circumference F. (iii) Use a straight edge to draw in the intervals EC and BF. Where they cross is the center of the circle.
A
B
D
C
2 Follow these steps to join two straight parallel line segments with a smooth, continuous curve. (i) Join BC with a straight line and then construct a perpendicular bisector to the line BC. Label the point of intersection M. (ii) Construct perpendicular bisectors to the sub intervals BM and CM. (iii) Construct a line perpendicular to AB, down from the point B to intersect with the perpendicular bisector of BM. (iv) Place compass point on the new intersection point and draw an arc from B to M. (v) Construct a line perpendicular to CD, up from the point C to intersect with the perpendicular bisector of CM. (vi) Place the compass point on the new intersection point and draw an arc from C to M.
A
* AWESOM
B
E
*
*
C
..../...
D
../20...
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Angle construction: Copying an angle Angles between 0° and 180° can be copied (or duplicated) using the following construction techniques. B
F
= C
A
1 Set up
D
E 2
• Draw an arc crossing both arms of angle A
• C opy this angle (or ‘construct a congruent angle’)
• Label the intersection points B and C • Draw a similar arc from D • Label the intersection point E
A Congruent means equal
B
• D raw a ray with the start point labeled (in this case, with D)
A
C
A
D
E
D
3
4
• S et the radius of the compass to the distance BC
• R ule a ray starting from point D through point F
• F rom point E, draw an arc across the one from step 2
B
• Label the point of intersection F A
B
F
C
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BAC =
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How does it work?
Your Turn
Constructions
Angle construction: Copying an angle Construct copies of each of these angles. Show all construct lines used. 1
A
E ANGL
OPYING A *C N
ANGLE * AN C
Use the exact same method when copying obtuse angles.
T TRUC CONS
G
2
* COP YI GLE N AN ION
0...
..../...../2
E
NG AN ANG YI L OP
A
3
For this diagram, construct
D
CDE as an exact copy of
BAD What geometrical statement can you make about the rays AB and DC following the construction of the angle CDE?
B
A
D
E
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Angle construction: Bisecting an angle Angles can be divided into two, equal smaller angles using these construction techniques. C
A
1 Set up
B 2
• W ithout changing the compass, draw an arc further out from point C
• C onstruct the bisector (a line that cuts it into 2 equal parts) of this angle
B
A • Draw an arc across both rays • Label the intersection points B and C
A
C
B
A
C
3
4
• Repeat step 2 with the compass at point B • Label the point of intersection D
• R ule a ray (the bisector) from the vertex A through the point D • Mark the equal angles with dots B
D
B A
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Bisector
C BAD =
C
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BAC = 50o ,
CAD BAD =
CAD = 25o
How does it work?
Your Turn
Constructions
Angle construction: Bisecting an angle
E* GL
CTING AN A SE N BI E
L ANG
(i)
XYZ (ii)
B
..
Y
Z
X C
2
A
Use the exact same method when bisecting obtuse angles. Construct the bisector EG for the obtuse DEF. Name the two equal angles formed by the bisector of DEF.
E D
3
...
/20 . . . . . ../
AN ANGLE NG * TI
CAB and (ii)
LE * BI SE ANG C
Construct the bisector for (i)
AN
O CTI U R ST CON
Bisect these angles. Show all construct lines used. 1
BISECTIN N G
F
Prove using construction methods, whether or not the ray QS bisects PQR below: Is the ray QS the bisector of
S
PQR?
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Constructions
Angle construction: 60o and 30o angles Angles of specific sizes can be constructed using a compass and straight edge.
60o
A
30o
A
60o
30o • Draw a large arc from point A
• Draw a large arc from point A
A
A
• W ith the compass on the intersection point, draw an equal sized arc that crosses the first one
• W ith the compass on the intersection point, draw an equal sized arc that crosses the first one A • F rom the new intersection point, draw an equal sized arc across the second one A
• D raw a line from A through the intersection of the two arcs • Label the angle 60o
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• R ule a line from A through the new intersection of the two arcs
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How does it work?
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Constructions
Angle construction: 60o and 30o angles Show all construct lines used. 1
Construct the following sized angles with the vertex at the start of the ray, X (ii) 30o
(i) 60o
X
X
2
(i) Construct a 60o angle with the vertex at the point A on the line segment below. (ii) Bisect this new angle to split it into two 30o angles.
GL AN
AND 30 O A NG E O
60
...
./20 . . . . / . ...
O 30 ANGLES * ND
3
B
LES * 60 O ANG A
A
O
CT RU T NS CO
0
60O A S * N ND 3 E L IO
(i) Construct a 30o angle with the vertex at point X on the line segment below. (ii) Construct a 60o angle with the vertex at point Y on the line segment below. (iii) Extend the two constructed arms until they meet to form a triangle at point Z. What type of triangle (ΔXYZ) has been formed by combining these angle constructions?
X
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Constructions
Angle construction: 45o and 90o angles Angles of specific sizes can be constructed using a compass and straight edge.
45o
45o
90o • W ith the compass at A, mark a dot O above the line
• C onstruct a perpendicular bisector to the line segment AB (go back to check if you forget how)
O
A
M
A
B
• F lip the compass around and draw a large arc from above O and through the line twice • S et the compass to the distance AM and draw an arc from A to the perpendicular bisector
• Label the second intersection point B
O
M
B
A
• R ule a line from A through the intersection of the arc and perpendicular bisector
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• Rule a line from C to A to make a right angle
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• R ule a line from B, through O until it crosses the arc again at C (BC is the diameter of the circle) • Mark a right angle at A
• Label the angle 45o
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Constructions
(ii) 90o
N
M
L 2
M
(i) Draw (or construct) an angle below that is exactly 45o at the end point A on the ray AB below. (ii) Copy the angle from part (i) at B to create a pair of parallel lines.
Rememb A
3
O
..../...../20.0 ..
Construct the following sized angles with the vertex at M. (i) 45o
ON CTI U R ST CON
O AND 90 O 45 A
1
ANGLES * 45
ANG
Show all construct lines used for these construction questions.
O
90
O A D 90 NGLE S AN
Angle construction: 45o and 90o angles
O S * 45 AND LE G N LE
Your Turn
*
How does it work?
er me?
B
(i) Construct a 90o angle with the vertex at point X on the line below. (ii) Construct a 45o angle with the vertex at point Y on the line below. (iii) Extend the two constructed arms until they meet (point Z) to form a right-angled triangle.
What is special about the right-angled triangle formed in this construction?
X
Y
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Constructions
Combining line and angle constructions Show all construct lines used for these trickier questions requiring combinations of construction techniques. Construct the following sized angles with the vertex at X. (i) 15o
(ii) 22.5o hint: this is half the size of which angle?
hint: this is half of 30
o
X
W 2
X
Y
Construct a 75o angle with the vertex at the point P on the ray below. hint: this is 15o less than a 90o angle or 30o more than a 45o angle.
D
LINE A N NG
ANGLE CO
OM * C BINI
1
20...
..../...../
S
TRUCTION NS
P 3
(i) Construct a perpendicular bisector to the interval XY. Label the intersection point O. (ii) Construct two 45o angles below the interval from each end point X and Y. Draw the arms until they intersect. (iii) Construct a 30o angle with the vertex at point X above interval XY. (iv) Construct a 60o angle with the vertex at point Y above interval XY. Draw the arm so it intersects with the 30o angle. (v) Set the compass to the radius OX and draw a complete circle.
X
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What is special about the intersection of the angles and the circle drawn?
Y
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Cyclic polygon construction: Regular hexagon inscribed in a circle A shape inscribed inside a circle is one where all the vertices touch the circumference of a circle. These shapes are called cyclic polygons.
1 Set up
2
• Draw a complete circle and label the center O
• W ithout changing the compass, place the point on the circumference and draw a small arc
O
O
3
4 • R ule a line from one arc intercept to the next to create a regular hexagon 25
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• M ove the compass point to the arc drawn, and repeat the same process all around the circle
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Constructions
Cyclic polygon construction: Regular hexagon inscribed in a circle Show all construct lines used for these. 1 (i) Construct a cyclic hexagon ABCDEF in the circle below using the point A the first vertex. (ii) Keeping the radius of the compass the same, put your compass point at each vertex of the hexagon and draw arcs starting and finishing on the circumference to produce a flower petal pattern.
OLYGON IC P CO CL CY
CIRCL IN A E
2
/20...
.. ..../...
D BE
O
(i) Construct a hexagon UVWXYZ in the circle with center O below using the point U as the first vertex. (ii) Construct a 30o angle using the side UV, with the vertex at U. Extend the newly constructed arm until it meets the circumference of the circle. (iii) Construct a 60o using the side WX, with the vertex at X. Extend the newly constructed arm all the way across the circle.
After the straight line UX is drawn in, what special type of triangle is formed?
O
U
22
AGON INSC HEX RI
A
AR
ON UCTI : REGU TR L NS
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Triangle construction: Equilateral triangles The basic construction techniques covered earlier can be applied to create specific shapes. Equilateral triangle (side length method)
Equilateral triangle (angle method) • Construct a 60o angle at A on the line segment
• S et the compass to the length of the interval AB
A
B
• Draw a large arc from one end (A) A
B
• Construct another 60o angle at B B
A
• Do the same from the other end of the line
A A
B
B
• R ule over the side constructions to the point of intersection to create an equilateral triangle
• Rule the sides from A and B to the point of intersection C C
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60o
60o
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Constructions
1
X 2
(i) Constructing the angles
TRIANGLE
Construct equilateral triangles on XY below using the method indicated.
(ii) Constructing the side lengths
X
Y
...
/20 . . . . /.
....
EQUILATER
Show all construct lines used for these triangle constructions.
N:
Triangle construction: Equilateral triangles
RU NST CTIO CO
TRIANGLE S
Your Turn
AL
Where does it work?
Y
The angle method can also be used to construct isosceles triangles. Construct an isosceles triangle using the given base AB, and with two equal angles of 30o
Remember: Isosceles triangles have two equal sides, opposite equal angles
A
3
B
base of isosceles triangle
(i) Create a rhombus JKLM by constructing equilateral triangles on either side of the line segment below. (ii) Construct the perpendicular bisector to the diagonal JL, ensuring the line passes through K and M. (iii) Use an angle construction technique to see if the perpendicular bisector of JL also bisects JKL. J
What geometric property of a rhombus has been shown by these constructions?
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Triangle construction: Median of a triangle A median is a line which divides the area of a triangle down the ‘middle’ into two equal halves. B Median of ΔABC
1
2 C A Area of triangle 1 = Area of triangle 2
1 Set up
2
• C onstruct the perpendicular bisector through AB
• C onstruct the median of ΔABC from the side AB through the vertex C
C
C
A
A
B
3
B
4
• M ark the midpoint of the line AB with a letter (D)
• Rule a line from D to the vertex C
• Mark the equal lengths AD and DB C
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A A
D
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1 0
B
Area of ΔACD = Area of ΔBCD
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Your Turn
Constructions
Triangle construction: Median, centroid and circumcenter Show all construct lines used for these triangle constructions. 1
Construct the median of the triangle below from the midpoint (M) of the side XY through the vertex W. X Name the two equal–sized triangles formed by your construction.
W Y
2
The centroid (O) is the point where all the medians drawn from each side of a triangle intersect. Find the centroid by constructing the median lines from every side of this triangle. GLE CON ST IAN TR
..../
.....
/20..
.
UMCENTER RC CI
B C
All the small triangles formed by the three median lines will have the exact same area!
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NTROID AND CE
A
,
ION: MEDIA CT N RU
Where does it work?
Your Turn
Constructions
Triangle construction: Median, centroid and circumcenter The circumcenter of a triangle is found using the same construction methods as those used for finding the centroid of a triangle. The only difference is that instead of drawing in the median lines, we look at where the perpendicular bisectors of each side cross each other. This point is the circumcenter of the triangle. 3 Follow these steps to construct the circumcenter for ΔJKL below. (i) Construct the perpendicular bisector for the side JK. (ii) Construct the perpendicular bisector for the side KL. (iii) Construct the perpendicular bisector for the side JL. (iv) Use a straight edge to extend the perpendicular bisectors to find the point they all intersect each other. Label the circumcenter (C). K
J
L
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Construct the circumcenter for ΔXYZ below. For this one, the circumcenter will be outside the triangle. Do the constructions and see for yourself!
Z
Y
X
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Where does it work?
Constructions
Triangle construction: Orthocenter of a triangle The orthocenter of a triangle is where the altitudes of a triangle all intersect each other. The altitude of a triangle is a line that passes through the vertex, perpendicular to the side opposite it. 1 Set up
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• Construct the orthocenter (O) of ΔPQR below
• C onstruct the line perpendicular to PR, passing through the vertex Q Q 0
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• R Q is extended (produced) to S with a dotted line to enable the following construction • C onstruct the line perpendicular to RQ, passing through the vertex P
• Produce PQ to T to enable next construction • C onstruct the line perpendicular to PQ, passing through the vertex R • Label the orthocenter (O)
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The correct term used to extend lines is produced. The order that the line segment is named is important. The line segment PQ is produced to S P 28
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The line segment QP is produced to S
S
S Constructions
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P
Q
Your Turn
Constructions
Triangle construction: Orthocenter of a triangle
TRIANGLE C
Construct the orthocenter (O) for these triangles below: 1
2
CENTE R THO OR
Show all construct lines used for these.
UCTION STR : N O
...
/20 . . . . /.
....
OF
Where does it work?
A TRIANGLE
B E
D
A C F
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You will need to produce the side IH to J for this one.
4
You will need to produce LM and KM. The perpendicular line for vertices K and L will join with these produced lines outside of the triangle. K
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L
M H
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Your Turn
Constructions
Triangle construction combo time: The Euler line Leonhard Euler (pronounced Oiler) was a Swiss mathematician who discovered that the orthocenter, centroid and circumcenter for any triangle are collinear. This means they all lie in a perfect straight line. Pretty cool! For the triangle below, construct the orthocenter, centroid and circumcenter and join them with a straight line to show Euler’s line.
X
Y
Z
C TRIANGLE
..../
EULER LIN E
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20. / . . . .. HE
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TIME: MBO T CO
UCTIO STR N N O
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Constructions
Circle construction: A circle that passes through three non-collinear points Non-collinear points are points that together, do not form a straight line. X Y
Z
1 Set up
2
• C onstruct a circle that passes through these three non-collinear points
• R ule two line segments between two pairs of points (eg PQ and PR) P 20
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• C onstruct the perpendicular bisector through both line segments
• S et the radius of the compass to the distance from O to any of the points P, Q or R
• Label their intersection point O
• W ith the compass at O, draw a full circle, passing through all the points
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Constructions
ON
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RUC TION
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POINTS * A
b
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RCL
..../..
R EA
For each group of three points below, construct the circle that passes through them.
CI
Show all construct lines used.
B
HROUGH S T TH SE
LE THAT PA RC S I C
ircle construction: A circle that passes through C three non-collinear points
O
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NON-COLLI N
Your Turn
E RE
What else can you do?
M
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A L
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Circumscribing a circle on a triangle means drawing a circle that touches each vertex of a triangle. So it is exactly the same method for construction, simply replacing the points with the vertices. Circumscribe a circle on each triangle below by following the same steps as a circle passing through three points. a
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Constructions
Circles: Tangents to a circle from an external point A tangent is a line that just touches the circumference of a circle at one point. Two or more lines that pass through the same point are called concurrent lines.
1 Set up
O
P
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• C onstruct the perpendicular bisector to find the midpoint (M) of the line segment OP
Construct tangent lines through the external point to the circle shown • Rule a line from the circle (O) to the point (P)
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• S et the radius of the compass to the distance OM
• R ule the tangents to the circle at A and B, that pass through the external point P
• Draw a circle with the compass at M • L abel the points it crosses the circumference A and B
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Your Turn
Constructions
Circles: Tangents to a circle from an external point Show all construct lines used. 1
C onstruct tangent lines that are concurrent (pass through the same point) with the given external point for the two circles below. b
a
O
G
O
I INT * CIR C PO
(i) Use a straight to draw a line segment QP that joins the center points. ..../ (ii) Use your construction skills to find the midpoint (M) of the line segment QP. (iii) Construct tangent lines from the point M to the circle with center P. (iv) Show that these tangents are also tangents to the circle with center Q by extending them.
AN EXTERNA L
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0. ... ../2 OM
The two circles below are the exact same size.
R A CI CLE F R
2
TO
TANGENT S S: E L
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Constructions
Circles: Incircle of a triangle Incircles are drawn inside a triangle with the circumference just touching each side once. 1 Set up
2
• B isect any two angles (in this case and BCA)
• Construct the incircle of the triangle ABC
BAC
• Extend the bisectors until they cross each other (O) A
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4 • W ith the compass set to the distance OR, draw a circle with the compass at point O
• C onstruct a line perpendicular to the side common to both angles (AC), passing through the point O
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Remember: Always leave the construct lines on your drawing
R Here is what the incircle of ΔABC looks like without the construct lines
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Your Turn
Constructions
Triangle constructions: Incircle of a triangle Show all construct lines used for these constructions. For ΔABC below: (i) Construct the bisector for BAC. (ii) Construct the bisector for ABC. Extend bisector to intersect with the bisector of (iii) Construct a line perpendicular to the side AB through the point D. (iv) Use your compass to draw the incircle of ABC with center at point D. A
BAC and label D.
...
A TRIANGL E OF *
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ES: INCI RCL RC CI L
A TRIANGL E OF * ES: INCI RCL RC CI L
E
Cheat Sheet
Constructions
Here is a summary of the important things to remember for constructions
Line construction: Perpendicular bisector
A
B
A A
B
A
M
A
B
B
B
Line construction: Perpendicular line at a point on the line
A
B
P
A
P
B
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P
B
P
A
B
P
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B
Line construction: Perpendicular line through a point external to the line P A
B
P
P
P A
B
A
P
A
B
B
A
B
Line construction: Parallel line through a point external to the line R
R P
A
P B
A
P B
Q
A
P
P
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Q
A
B
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Q
B
Angle construction: Copying an angle B
B A
A
A
C
B A
C
C
F
F
D D
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Cheat Sheet
Constructions
Angle construction: Bisecting an angle B
B
B
B A
A
A
C
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Angle construction: 60o and 30o angles
60o 60o
A
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30o A
30o
A
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Angle construction: 45o and 90o angles
45o o
A
M
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B
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B D
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90o A
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Cyclic polygon construction: Regular hexagon inscribed in a circle
O
O
O
O
Triangle construction: Equilateral triangles Side length method Angle method
C
60o
A
B
A
B
A
B
A
B
A
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B
A
B
A
60o
60o
B
Cheat Sheet
Constructions
Triangle construction: Median of a triangle C
C
C
C
C B
A
B
A A
A
A
B
B
D
B
Triangle construction: Orthocenter of a triangle Q
Q
Q
Q P
P
R
P
O
R
R
P
R
Circle construction: A circle that passes through three non-collinear points P
P
P P
R
R
R R
Q
Q
Q
O
Q
O
Circles: Tangents to a circle from an external point
A O
P
O
O
P
M
M
P
O
P
M B
Circles: Incircle of a triangle B
A A
B
B
A
O
O
R A C
B
C
O
A
R
B
C
O
R C
C
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Constructions
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Notes
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Constructions
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