Download Skills: To construct circles of a given radius, angle bisectors, and

Constructions: angle bisector
Skills: To construct circles of a given radius,
angle bisectors, and equilateral triangles.
Constructions are done only using a compass and a straight edge.
A compass draws part of a circle - an arc
Leave all the construction lines - they are the “working”
Circles – Use a ruler to set the compass to the radius of the circle
Angle bisector
Definition:
eg) Angle bisect angle FGH
F
G
F
G
H
H
Constructing
Startertriangles A
1) Construct an equilateral triangle with sides of 7cm. Label it ABC
2) Angle bisect angle ABC. Continue the angle bisector until it touches line AC.
Label this point P
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0
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3) Angle bisect angle APB and CPB.
Continue the angle bisectors until
they touches the sides of the triangle.
Label these points Q and R
4) Measure the distance
QR in mm. QR = ______ mm
0
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Skills: Construct triangles given sides and angles.
Given 3 sides lengths
1) Rule and measure the longest side
2) Set compass length of second side and arc.
3) Set compass to the length
of the third side and arc again
so the arc’s cross. Mark this point
4) Rule in the triangle sides
Construct a triangle with side
lengths 3cm, 4cm and 5cm
4cm
5cm
3cm
0
1
2
3
4
5
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5
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3
2
0
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Constructing triangles B
Given 2 sides and 1 angle
1) Rule the longest side (correct length)
0
2) Measure the angle with a protractor
3) Rule the second side (correct length)
4) Rule in the third side
Construct a triangle with side
lengths 5cm, 4cm and a
45° angle between them
4cm
45°
5cm
1
2
3
4
5
0
1
2
3
4
5
Constructing triangles C
Given 1 side and 2 angles
1) Rule and measure the side length
0
1
2
3
4
5
2) Measure the angle to each of the other sides.
Rule the sides in until they cross.
Eg: 1) Construct a triangle with side length
BC 5cm, and a 40° and 50° angles
2) Measure the sides AB = ____
AC = ____
3) Measure the angle BAC = ____
Eg: 1) Construct a triangle with side length PQ 3cm,
and a 30° and 120° angles
2) Measure the sides PR = ___
R
QR = ____
3) Measure the angle PRQ = ____
30°
P
120°
3cm
Q
Perpendicular bisector
Skill: Construct perpendicular bisectors.
Definition: ___________________________________
B
A
Construct the perpendicular bisector to the line AB
B
A
Note that line AB is now cut in half at 90°
B
A
Perpendicular line from point near a line
Skills: Construct perpendicular lines from a point near a line.
Construct a line passing through point P
which is also perpendicular the line DE
P
.
E
D
Put compass point at P
Adjust the compass to reach just over
line DE so the arc cuts the line twice
Mark these points A and B
P
Now construct the perpendicular bisector
of line AB as before.
Note that line DE is now cut at 90°
D
.
E
Perpendicular line from point on a line
Steps: Construct perpendicular lines from a point on a line.
Construct a line passing through point P
which is also perpendicular the line GH
H
P
.
G
Put compass point at P
Adjust the compass so the arc cuts the
line GH twice.
Mark these points A and B
Now construct the perpendicular bisector
of line AB as before.
Note that line GH is now crossed at 90°
H
P
G
.
Find the circumcircle
Given any triangle construct a circle which will pass through all three corners of
the triangle.
Steps:
1) Rule a neat large triangle.
2) Perpendicular bisect all three sides
3) Continue the perpendicular bisectors
until they intersect. Label this point P
4) Put compass point at P and construct a
circle passing through the corners of the circle.
Extension: Circumcircle
Given any triangle construct a circle which will pass through all three corners of
the triangle.
Extension: Centre of gravity
Given any triangle use construction to find the centre of gravity of the triangle.
1) Rule a neat large triangle.
2) Perpendicular bisect all three sides. Don’t rule in the perpendicular bisector,
just mark the midpoint of the side.
3) Rule a line from a corner of the triangle to the middle of the opposite side
found in step 2).
4) Repeat step 3) for the other two corners of the triangle.
5) Where the three lines meet mark point P. This is the centre of gravity (balance
point of the triangle)
6) Repeat this on a piece of scrap paper. Carefully cut out the triangle. Put a
pen point on point at P see if the triangle balances.
Extension: The “altitude” is the line from a corner of a triangle which intersects
the opposite side at 90° Do all 3 altitudes of a triangle meet?
Construct a Square
1) Construct a circle. Mark the centre C
2) Use a ruler to mark two opposite
points on the circle edge (mark A & B)
Check AB passes through C
C
3) Perpendicular bisect line AB
.
4) Where the perpendicular bisector
crosses the circle mark D & E
5) Join ADEB to make a square
Extension: Can you find a different way to make a square only using a compass
and straight edge?
Construct an octagon
Can you now make an
octagon (8 sides) using the
square as a starting point?
Parallel Lines
ARC METHOD
1) Put compass point at P Adjust the compass to reach JUST to line AB Keep the
compass the same.
2) Now put the compass point
on line AB and arc above
the line. Repeat several times
3) Rule a line connecting
the tops of the arc’s
P
.
A
Can you find another different way to construct parallel lines?
Gamma p151
p153 Ex 9.1
Hw
p71 Ex 17.01
B
RHOMBUS METHOD
1) Put compass point at P Adjust the
compass to reach OVER to line AB
Mark this point E
Keep the compass the same
for all steps.
P
A
.
2) Put compass point at E Arc across
line AB again. Mark this point F
B
3) Put compass point at P and arc into
“space” roughly parallel to line AB
A
4) Put compass point at F and arc
To cross the arc in step 3) Mark this point Q
P
.
5) Rule a line through P and Q.
This should be parallel to AB
B
Construct a Hexagon
1) Construct a circle
Keep the compass the same.
2) Mark point P anywhere
on the circle. Put compass
point on P and arc on
the circle point Q
3) Put compass point on Q
and arc on the circle again.
4) Repeat until you go around the circle
5) Join the arc points to form a hexagon.
Construct a dodecagon
Can you now make an
dodecagon (12 sides) using the
hexagon as a starting construction?
C
.
Extension: A pentagon
1) Construct a circle. Mark the centre C
2) Use a ruler to mark two opposite points on the circle edge (mark A & B)
Check AB passes through C
3) Perpendicular bisect line AB. Where the perpendicular bisector crosses the
circle mark D & E
4) Find the midpoint of BC, Label it N
5) Put the compass point on N. Adjust the compass
to reach C. Construct a circle
6) Find the midpoint of AC, Label it M
7) Put the compass point on M.
Adjust the compass to reach C.
C.
Construct a circle
8) Put the compass point on D.
Adjust the compass just reach the
circles in step 5) and 7) Construct a circle
9) Mark the two points where this circle cuts
the original circle. Adjust the compass to this
length. Keep the compass the same.
10) Arc around the original circle using this compass setting.
Joining these points should form a regular pentagon
NAME _______________________________
Angle bisect these angles
Perpendicular bisect these lines