## Constructions

### Types of Diagrams

 A sketch is a rough diagram which does not have to be too accurate.A drawing is a more accurate and can be drawn using a ruler and a protractor for measuring lengths and angles. A construction in mathematics is an accurate drawing done usually using only instruments such as a ruler as a straight edge, and a pair of compasses. When doing a construction, leave all of your construction arcs and lines on the diagram.In year 9, topic 26 covered the construction of circles and triangles.This topic covers more advanced constructions. ### Perpendicular bisector

perpendicular bisector, sometimes called a mediator, is a line which cuts another line in half and at right angles.

 To bisect AB at right angles: Step 1 Draw equal arcs, centres A and B, of radius more than half of AB to meet at P and Q. Step 2 Join P and Q. PQ is the perpendicular bisector of line AB ### Angle Bisector

An angle bisector, cuts or divides and angle in half.

 To bisect angle ABC: Step 1 Draw an arc, centre B, to cut AB at P, and BC at Q. Step 2 Draw equal arcs, centres P and Q, to intersect at R. Step 3 Join B to R. BR is the angle bisector of ∠'ABC ### Parallel lines

 To construct a line parallel to AB passing through C. Step 1 Mark a point Q on the line AB. Step 2 Draw an arc, centre Q, radius QC, to cut AB at P. Step 3 Draw arcs with same radius, centres C and P, to intersect at R. Step 4 Join C and R. CR is parallel to AB ### Perpendicular lines: Method A

 To construct a line perpendicular to AB passing through C: Step 1 Draw arc, centre C, to cut AB twice at P and Q. Step 2 Draw equal arcs, centres P and Q, to meet at R. Step 3 Join C and R. CR is perpendicular to AB ### Perpendicular lines: Method B

 To construct a line perpendicular to AB passing through C: Step 1 Draw equal arcs, centre C, to cut AB twice atP and Q. Step 2 Draw equal arcs, centres P and Q, to meet at R. Step 3 Join C and R. CR is perpendicular to AB ### Constructions from Given Data

Some shapes and figures can be drawn from given data such as lengths and angles using ruler and protractor .

Equilateral Triangle

Construct an equilateral triangle of side length 5 cm

 Step 1 Draw an arc, length 5 cm, centre A Step 2 Draw an arc from a point B, to cut the original arc at C. Step 3 Join ABC ABC is an equilateral triangle Draw a quadrilateral ABCD, with angle A = 110° and angle B = 80° and sides of AD = 4 cm, BC = 5 cm and base AB = 6 cm. Step 1 Draw the line AB of length 6 cm. Step 2 Set compasses to 4 cm and draw arc on A. Step 3 Set compasses to 5 cm and draw arc on B. Step 4 Use protractor to draw angles of 110° and 80° on A and B Step 5 Join the four points. Hexagon

 Construct a regular hexagon Step 1 Draw a circle Step 2 Set compasses to 5 cm and draw arc on A. Step 3 Continue working round circle making arcs of 4 cm. Step 4 Join each point of intercection of the arcs with the circle together. Triangles

Two constructions in triangles that are often required are:

The medians of a triangle are lines from each vertex to the middle of the opposite side.

The altitudes of a triangle are lines from each vertex which meet the opposite side of the triangle at right angles.