Download Loci and Constructions

WJEC MATHEMATICS
INTERMEDIATE
ANGLES AND GEOMETRY
CONSTRUCTIONS AND LOCI
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Contents
Perpendicular Bisector of a Line
Angle Bisector
Drawing an angle of 60o
Drawing an angle of 30o
Constructing a Triangle - 3 Lengths
Constructing a Triangle - 2 Sides, 1 Angle
Constructing a Triangle - 1 Side, 2 Angles
Locus
Locus Example
Credits
WJEC Question bank
http://www.wjec.co.uk/question-bank/question-search.html
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Perpendicular Bisector of a Line
Bisecting a line means to divide it evenly in two.
Example
Begin with a line,
A
B
Step 1
Place your compass at point A, open it over halfway and draw an arc
A
B
Step 2
Without altering the compass, place the compass at point B and
draw another arc
A
B
This has
created an
angle of 90o
Step 3
Draw a line through both points of intersection.
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Angle Bisector
Bisecting an angle means dividing (splitting) the angle evenly in two
Example
Begin with an angle
B
A
C
Step 1
Place your compass at point A and draw an arc that crosses both
lines
B
A
C
Step 2
Without changing the compass, place the compass at the two points
of intersection and create two arcs that overlap
B
A
C
Step 3
Draw a line that originates at point A and goes through the two arcs
that you have just drawn
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Drawing an angle of 60o
Begin with a line
A
B
A
B
A
B
Step 1
Place your compass point on A and open the compass to the same
length as the line AB. Draw an arc.
Step 2
Without altering the compass, place the point on B and draw another
arc.
Step 3
Connect either point A or B to the point where the two arcs meet.
This will give you an angle of 60o
60o
A
B
If you connect point B to the intersection you will have an equilateral
triangle.
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Constructing an angle of 30o
To construct an angle of 30o, construct a 60o angle and then bisect
that angle as seen previously in this booklet.
Practice drawing angles of 60o and 30o
Exercise G19
1. Bisect the following lines
2. Bisect the following angles
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Constructing a Triangle - Knowing All Lengths
Construct an accurate drawing of a triangle with side lengths 10cm,
6cm, and 5cm.
Step 1
Use a ruler to draw a line the length of a side given in the question.
(10cm)
10cm
Step 2
Open the compass to the length of the second side (6cm) and draw
an arc from one end point
10cm
Step 3
Open the compass to the length of the third side (5cm) and draw an
arc from the other end point.
5cm
6cm
10cm
Step 4
Join the end points to the intersection to create the triangle
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Constructing a Triangle - Knowing 2 Sides, 1 Angle
Construct a triangle with a length of 4.5cm, a length of 6.6cm, and an
angle of 47 degrees.
Step 1
Start by drawing a line equal to a length given in the question.
6.6cm
Step 2
Place your protractor on one end of the line and make a mark at 47o
x
6.6cm
Step 3
Draw a line, the length given in the question (4.5cm) through the
mark made in step 2
4.5cm
x
47o
6.6cm
Step 4
Join the two lines to make a triangle
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Constructing a Triangle - Knowing 2 angles, 1 side
Construct an accurate diagram of a triangle with a side length of
5cm, an angle of 45o, and an angle of 58o
Step 1
Begin by drawing a line the length given in the question (5cm)
5cm
Step 2
Place your protractor on one end and measure an angle given in the
question (45o). Mark this with a cross.
x
5cm
Step 3
Place your protractor on the other side and measure the other angle
given in the question (58o). Mark this angle with a cross.
x
x
Step 4
5cm
Draw a line from either end through the points made
x
x
5cm
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Exercise G20
1. Construct a triangle with side lengths 12cm, 7cm, and 8cm
2. Construct a triangle with side lengths 15cm, 9cm, and 6cm
3. Construct a triangle with side lengths 10cm, 9cm, and 5cm
4. Construct a triangle with a side length 12cm, a side length 3cm,
and an angle of 65o
5. Construct a triangle with a side length 8cm, a side length 6cm,
and an angle of 73o
6. Construct a triangle with a side length 15cm, a side length 5cm,
and an angle of 44o
7. Construct a triangle with a side length 10.5cm, an angle 49o, and
an angle of 86o
8. Construct a triangle with a side length 8.6cm, an angle 57o, and an
angle of 71o
9. Construct a triangle with a side length 4.3cm, an angle 33o, and an
angle of 36o
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Locus
A locus is a path. A locus around a point is the path that remains a
constant distance from the line at all times.
Example 1
The locus of
all points
3cm from A
The locus of
A
all points
3cm
less than
3cm from A
Example 2
Shade the region that is closer to point A than point B
A
x
x
B
Imagine AB
is a line, we
know how to
bisect this
A
x
x
B
This shaded
region
contains all
points closer
to A than B
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Example 3
Shade the region that is closer to the line AB than the line AC
B
C
A
Begin by bisecting the angle
B
C
A
Now, shade all points closer to AB
A
B
C
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Example 4
Sketch the line that remains 5cm from the line AB
Use a compass on
either end and a ruler
in the middle
Example 5
Shade the region of points that satisfy the following conditions;
 Closer to AC than AB
 Less than 4cm from B
Bisect the angle at A
to see which side is
Using a compass to
closer to AC than AB
draw a line 4cm from
B
Region that satisfies
both conditions
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Exam Questions G29
1.
2.
14
3.
4.
15
5.
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