Maths- Constructions, Loci, Trigonometry and Maps Flashcards
Perpendicular bisector of a line segment
The locus of a point which moves so that it is an equal distance from two points, A and B, is the perpendicular bisector of the line joining A and B.
Perpendicular means at right angles to.
Bisector means cuts in half.
To construct this locus, you do the following (try this yourself on a piece of paper):
To bisect an angle, you do the following:
Place your compass on V and draw an arc that crosses both sides of the angle.
Label the crossing points A and B.
Place your compass on A and draw an arc between the two sides of the angle.
Without adjusting your compass place it on B and draw another arc that cuts the one you just drew. Label the point where they cross C.
Draw a straight line through V and C.
Remember:
if you are asked to do a construction in an exam, do not rub out your construction lines.
Loci:
A locus is a path. The path is formed by a point which moves according to some rule.
The plural of locus is loci.
Example
The locus of a point moves so that it is always a set distance (x) from a fixed point (O). What shape is it?
Imagine the minute hand on a clock. As the hand moves around the clock face, think of the path it follows.
Remember: the easiest way to draw this locus around a set point is to use a compass.
Example 2:
The locus of a point moves so that it is always a set distance (r) from the line between the points P and Q.
What shape is it?
The straight lines of the locus are parallel to the line from P to Q, because they are at a set distance (r) from the line. P and Q are fixed points at either end of the line, so we draw semicircles of radius r.
Sometimes the locus is not just a line, but an area. For example:
A cow grazing in the field ABCD moves so that it is always a distance of 5m from fence AB. Draw the locus of the cow.
Drawing triangles
There are two methods of drawing triangles - construction or using a ruler and protractor.
Construct a triangle with side lengths of 6cm, 5cm and 4cm.
Solution
Use a ruler to draw a 6cm line. Label one end A and the other B.
Open the compass to a radius of 5cm.
Place the compass needle at point A and draw an arc above the line.
Open the compass to a radius of 4cm.
Move the compass needle to point B and draw an arc above it.
Join each end of the line to the point where the arcs cross.
Remember, do not erase any construction arcs when using this method.
Trigonometry
Trigonometry can be used to calculate the lengths of sides and sizes of angles in right-angled triangles.
The three formulae
sin, cos, tan
The sides of the right-angled triangles are given special names - the hypotenuse, the opposite and the adjacent.
The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides relate to the angle under consideration.
There are three formulae involved in trigonometry:
sin = opposite / hypotenuse cos = adjacent / hypotenuse tan = opposite / adjacent
Which formula you use will depend on the information given in the question.
There are a couple of ways to help you remember which formula to use. Remember SOHCAHTOA
Map scales
Maps scales can be written in ratios and tell you how many units of land, sea etc are equal to one unit on the map.
If you are travelling from Manchester to Newcastle, for example, and need to know how far it is, it would be very difficult to work this out if the map does not have a scale.
Example
The scale of a map is 1:50 000. A distance is measured as 3cm on the map.
How many cm, m and km is this equivalent to in real life?
1 cm on the map represents 50 000cm. Therefore, 3cm on the map represents 150 000cm. To convert from cm to m, divide by 100. 150 000cm ÷ 100 = 1500m To convert from m to km divide by 1000. 1500m ÷ 1000 = 1.5km
