Constructions

GCSELevel 4-5WJEC

Constructions Revision

Constructions

Construction is the process of drawing accurate diagrams of shapes, angles and lines. These are called constructions.

It is important to be as accurate as possible, drawing lengths accurate to $2\text{ mm}$ , and angles accurate to $2\degree$ . You will need to be able to accurately use a ruler, pair of compasses, and protractor.

Construction 1 – Bisecting a Line

Bisecting means dividing something into two equal parts. So, when bisecting a line, we want to split the line into two lines of equal length.

The bisector of a line will be perpendicular to the original line.

Process:

1. On the line, for example line $AB$ here, start by setting your pair of compasses to a length that is over half the length of $AB$ . For example, if the line was $4\text{ cm}$ , your compasses would need to be set at over $2\text{ cm}$ .

Place your pair of compasses at point $A$ and draw an arc crossing the line.

2. Draw another similar arc with your compasses set at point $B$ , keeping your compasses at the same length.

3. Join the two points where the arcs cross with a straight line (drawn with a ruler). This line bisects $AB$ into two equal length lines.

Construction 2 – Bisecting an Angle

When we bisect an angle, we are again trying to split the angle in two equal halves. For example, if an angle was $60\degree$ , we would bisect it into two angles of $30\degree$ each.

Bisecting an angle means we are constructing a line equidistant from the two lines forming an angle.

Process:

1. To bisect the angle between the lines $AB$ and $BC$ , start by placing your pair of compasses at $B$ and drawing a small arc that crosses both $AB$ and $BC$

2. Place your pair of compasses at the point where your previous arc crosses the line $AB$ , and draw a small arc between the two straight lines. Repeat this at the crossing point on $BC$ , making sure the arcs cross over.

3. Using a ruler, draw a straight line from $B$ , that passes through the point were the middle arcs cross. This is the angle bisector.

Construction 3 – Point to Line

Another construction you may be asked to do is constructing the perpendicular line from a point to a line.

This is where you have a point, and a line, as below, and need to draw a line between them that is perpendicular to the original line.

Process:

1. With your pair of compasses at the point, draw two arcs, at the points where they cross the line.

2. Put your pair of compasses at one of the points whether the arc crosses the line, and draw an arc below the line. Repeat this from the other crossing point, ensuring the new arcs cross over.

3. With a straight line, draw a line from the point that cross the point of intersection between the arcs. This line will be perpendicular to the original line.

Construction 4 – Circles

You may be required to draw a circle using a pair of compasses.

To do this, set your compasses to the length of the radius of your desired circle, and place it where you would like the centre of the circle to be ( $0$ ). Draw the full circumference of the circle until your arc reaches the start of your arc.

Make sure to draw lightly to avoid the pair of compasses moving.

Construction 5 – Triangles

You may be asked to construct some triangles when given at least $3$ pieces of information about the triangle.

You will need one of the following options of information:

Side Angle Side (SAS)
Angle Side Angle (ASA)
Side Side Side (SSS)

Process – SAS

Example – construct a triangle with two sides of lengths $5\text{ cm}$ and $6\text{ cm}$ , with an angle of $50\degree$ between them.

1. With a ruler, draw the longest side

2. Using a protractor, measure the angle $50\degree$ and draw a dot to show this angle ( $x$ )

3. Draw a line of $5\text{cm}$ crossing, reaching, or going through, the point $x$ (depending on how far the point $x$ is), but extending your ruler through $x$ to ensure the angle is correct.

4. Finally, join up these two lines with a third line. This is the final triangle.

Process – ASA

Example – construct a triangle with two angles $40\degree$ and $65\degree$ , with a side of length $7\text{ cm}$ between them.

1. With a ruler, draw the side

2. Using a protractor, measure the angle $40\degree$ from one side of the line and draw a dot to show this angle ( $x$ )

3. Draw a long line through $x$

4. From the other side of the line, use a protractor to measure $65\degree$ , and draw a dot showing this angle ( $y$ )

5. Draw a line through $y$ , to the point it reaches the other line.

Process – SSS

Example – construct a triangle with sides of lengths $6\text{ cm}$ , $3\text{ cm}$ , and $5\text{ cm}$ .

1. With a ruler, draw the longest side, $6\text{ cm}$

2. Set your pair of compasses to one of the other lengths, for example $3\text{ cm}$ and draw an arc from one of the ends of the $6\text{ cm}$ line.

3. Repeat with your pair of compasses at $5\text{ cm}$ , with centre at the other side of the $6\text{ cm}$ line.

4. Draw lines from either end of the $6\text{ cm}$ line, joining at the point of intersection between the two arcs.

Construction 6 – Quadrilaterals

To construct quadrilaterals, we split them into $2$ triangles, and use the methods we learnt for constructing triangles to construct the quadrilateral.

To construct a quadrilateral, you need $5$ pieces of information. For example, side lengths, angle sizes, or the length of the diagonal through the quadrilateral.

For example, you might be asked to construct a quadrilateral with sides of lengths $AB=7\text{ cm}$ , $BC=8\text{ cm}$ , $CD=7\text{ cm}$ , and $DA=5\text{ cm}$ , with a diagonal length $AC=10.5\text{cm}$ .

This can be split into two triangles, and then the triangles can be drawn using the SSS method described earlier: