1

WJEC MATHEMATICS

INTERMEDIATE

ANGLES AND GEOMETRY

CONSTRUCTIONS AND LOCI

2

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

3

Perpendicular Bisector of a Line

Bisecting a line means to divide it evenly in two.

Example

Begin with a line,

Step 1

Place your compass at point A, open it over halfway and draw an arc

Step 2

Without altering the compass, place the compass at point B and

draw another arc

Step 3

Draw a line through both points of intersection.

A B

A B

This has

created an

angle of 90o

A B

4

Angle Bisector

Bisecting an angle means dividing (splitting) the angle evenly in two

Example

Begin with an angle

Step 1

Place your compass at point A and draw an arc that crosses both

lines

Step 2

Without changing the compass, place the compass at the two points

of intersection and create two arcs that overlap

Step 3

Draw a line that originates at point A and goes through the two arcs

that you have just drawn

A C

B

A C

B

A C

B

5

Drawing an angle of 60o

Begin with a line

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

If you connect point B to the intersection you will have an equilateral

triangle.

A B

A B

A B

A B 60o

6

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

7

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)

Step 2

Open the compass to the length of the second side (6cm) and draw

an arc from one end point

Step 3

Open the compass to the length of the third side (5cm) and draw an

arc from the other end point.

Step 4

Join the end points to the intersection to create the triangle

10cm

10cm

10cm

6cm 5cm

8

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.

Step 2

Place your protractor on one end of the line and make a mark at 47o

Step 3

Draw a line, the length given in the question (4.5cm) through the

mark made in step 2

Step 4

Join the two lines to make a triangle

6.6cm

6.6cm

x

6.6cm

x 4.5cm 47o

9

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)

Step 2

Place your protractor on one end and measure an angle given in the

question (45o). Mark this with a cross.

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.

Step 4

Draw a line from either end through the points made

x

x x

x x

5cm

5cm

5cm

5cm

10

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

11

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

Example 2

Shade the region that is closer to point A than point B

The locus of

all points

3cm from A

The locus of

all points

less than

3cm from A

A 3cm

A B x x

Imagine AB

is a line, we

know how to

bisect this

A B x x

This shaded

region

contains all

points closer

to A than B

12

Example 3

Shade the region that is closer to the line AB than the line AC

Begin by bisecting the angle

Now, shade all points closer to AB

A

B

C

A

B

C

A

B

C

13

Example 4

Sketch the line that remains 5cm from the line AB

Example 5

Shade the region of points that satisfy the following conditions;

Closer to AC than AB

Less than 4cm from B

Use a compass on

either end and a ruler

in the middle

Using a compass to

draw a line 4cm from

B

Bisect the angle at A

to see which side is

closer to AC than AB

Region that satisfies

both conditions

14

Exam Questions G29

1.

2.

15

3.

4.

16

5.