Bearings, Scale Drawing, Constructions & Loci (AQA GCSE Maths)
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What is a bearing?
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What is a bearing?
A bearing is a way of describing and using a direction as an angle.
From which compass direction should a bearing always be measured?
A bearing should always be measured from North.
True or False?
Bearings are measured in an anti-clockwise direction.
False.
Bearings are measured in a clockwise direction.
How many figures must a bearing be written with?
A bearing must be written with 3 figures.
For angles under 100°, zero(es) should be used to fill in the missing figures, e.g. 059º, 008º.
True or False?
SOHCAHTOA and Pythagoras' theorem are frequently used in bearings questions.
True.
Pythagoras' theorem and SOHCAHTOA are frequently used in bearings questions.
This is because situations often include finding missing lengths or angles in right-angled triangles.
If a bearing of A from B is known, how do you find the bearing of B from A?
If a bearing of A from B is less than 180o, then add 180o to find the bearing of B from A.
E.g. if the bearing of A from B is 030o, then the bearing of B from A is 210o.If a bearing of A from B is greater than 180o, then subtract 180o to find the bearing of B from A.
E.g. if the bearing of A from B is 270o, the bearing of B from A is 090o.Which compass direction is indicated by a bearing of 090°?
When a bearing is 090°, it means due east.
True or False?
North should always be indicated on a diagram.
True.
North should always be indicated on a diagram as a point of reference.
What does it mean when a scale for a diagram is given as a ratio?
E.g. a scale of 1 : 8.
When the scale is given as a ratio (e.g., 1:8), it means that 1 unit on the diagram represents 8 of the same type of units in real life.
E.g. 1 cm on map = 8 cm in real life
True or False?
Scales on maps are always given with units.
False.
Scales on maps are usually given in the form 1 : n with no reference to units.
True or False?
To convert from a real-life distance to a scale distance, you usually need to divide by the scale factor.
True or False?
To convert from a real-life distance to a scale distance, you usually need to divide by the scale factor.
How is a scale length on a map converted to a real-life length?
E.g. a length of 3 cm on a map with scale 1 : 20 000.
To convert from a scale length to a real length, multiply by the scale factor.
E.g. 3 × 20 000 = 60 000 cm (600 m).
How is a real-life length converted to a scale length on a map?
E.g. a real-length of 15 km using a map with scale 1 : 50 000.
To convert from a real length to a scale length, divide by the scale factor.
E.g. 15 ÷ 50 000 = 0.000 3 km (30 cm).
True or False?
It may be helpful to convert the scale to more suitable units when dealing with very small or large scales.
True.
It may be helpful to convert the scale to more suitable units when dealing with very small or large scales.
For example, if 1 cm on a map represents 25 000 cm in real life, it may be easier to convert the units and use 1 cm represents 250 m or 0.25 km.
What is the compass direction for a bearing of 270º?
The compass direction for a bearing of 270º is due west.
What is the bearing for the compass direction of due northeast?
The bearing for the compass direction of due northeast is 045º.
Remember to include 3 figures in your bearing.
True or False?
Scales on scale drawings are never given with units.
False.
Scales on scale drawings are often given as a description with units.
This is unlike maps, where the scales are often given as ratios without units.
After measuring a distance on a scale drawing, what is the next step to find the corresponding real-life distance?
After measuring a distance on a scale drawing, the next step to find the corresponding real-life distance is to multiply by the scale factor.
What equipment might you need to be able to construct a triangle?
The equipment you may need to be able to construct a triangle is:
A sharp pencil
A ruler
A protractor
A pair of compasses
You could be asked to construct a triangle for any of the following cases: SSS, SAS, or ASA.
What do each of these mean?
When constructing a triangle, it could be for any of the following cases:
SSS - you are given the lengths of all three sides.
SAS - you are given the lengths of two sides and the angle in between them (the included angle).
ASA - you are given the size of two angles and the length of the side in between them (the included side).
How would you construct a triangle when told the lengths of three sides? (SSS)
To construct a triangle when told the lengths of three sides (SSS):
Draw the longest side using a ruler as a horizontal base.
Use compasses set to the length of the next side and draw an arc from the end of the base line.
Repeat this for the third side, drawing an arc from the other end of the base line, the same length as the third side.
The arcs should intersect. Join the intersection of the arcs to each end of the base line using a ruler.
How would you construct a triangle when told the lengths of two sides, and the angle between them? (SAS)
To construct a triangle when told the lengths of two sides and the angle between them (SAS):
Draw the longest side using a ruler as a horizontal base.
Use a protractor to measure the given angle from one end of the base line and make a mark on the paper at this angle.
Use a ruler to draw the second given side length from the end of the base line through the mark you have just made.
Check the lengths and angles are all correct so far using a ruler and protractor.
Use a ruler to draw the remaining side, between the end of the base line and the end of the second given side.
How would you construct a triangle when told two angles, and the length of a side between them? (ASA)
To construct a triangle when told two angles, and the length of a side between them (ASA):
Use a ruler to draw the given length as a horizontal base.
Use a protractor to measure the first given angle from one end of the base line and make a mark on the paper at this angle.
Use a ruler to join the end of the horizontal base line to the mark you have just made, and extend the line further.
Repeat steps 2 & 3 for the second given angle, at the other end of the base line.
The lines should intersect, forming a triangle.
True or False?
You should leave all your working and construction lines on the page, and not rub any out.
True.
You should leave all your working and construction lines on the page, and not rub any out.
What is a perpendicular bisector?
A perpendicular bisector is a line that cuts another line exactly in half (bisects) and also crosses it at a right angle (perpendicular).
How would you construct a perpendicular bisector?
To construct a perpendicular bisector:
Set the distance between the point of the compasses and the pencil to be more than half the length of the line.
Place the point of the compasses on one end of the line and sketch an arc above and below the line.
Keeping your compasses set to the same distance, move the point of the compasses to the other end of the line and sketch an arc above and below it again.
The arcs should intersect each other both above and below the line, connect the points where the arcs intersect using a ruler.
True or False?
The shortest distance from a point to a line is a straight line, which meets the line at 90 degrees.
True.
The shortest distance from a point to a line is a straight line, which meets the line at 90 degrees.
E.g. When finding the shortest distance between a boat and a coastline, construct a perpendicular line from the boat to the coastline.
How would you construct a perpendicular to a line, from a point?
To construct a perpendicular to a line, from a point:
Set the distance between the point of your compasses and the pencil to be greater than the distance between the point and the line.
Place the point of the compasses onto the point and draw an arc that intersects the line in two places.
Make sure that the distance between the point of the compasses and the pencil is greater than half the distance between the intersection points. Place the point of the compasses on a point of intersection, and sketch an arc above and below the line.
Keeping the distance between the point on the compasses and the pencil the same, do this again from the other point of intersection. These should intersect with the previous arcs.
Join the original point with the point where the arcs intersect.
What is an angle bisector?
An angle bisector is a line that cuts an angle exactly in half (bisect).
How would you construct an angle bisector?
To construct an angle bisector:
Place the point of the compasses at the point of the angle and sketch an arc that intersects both of the lines that form the angle.
Set the distance between the point of your compasses and the pencil to be greater than half the distance between the intersection points. Place the point of the compasses on one of the points of intersection and sketch an arc.
Keeping the distance between the point of the compasses and the pencil the same, place the point of the compasses on the other point of intersection and sketch another arc, this should intersect the arc sketched in step 2.
Join the point of the angle to the point of intersection of the arcs with a ruler.
What shape is formed by points that are a fixed distance from a point?
The shape that is formed by points that are a fixed distance from a point is a circle.
True or False?
A path where all points are equidistant from a straight line will be a parallel line.
False.
A path where all points are equidistant from a straight line will not be a parallel line on its own.
A race track shape will be formed consisting of a parallel line on either side of the straight line and a semi-circle at either end of the line connecting the parallel lines.
What construction should be drawn when a question asks you to identify an area that is closer to one point than another?
To identify an area that is closer to one point than another, you must find the path where all points are equidistant from the original two points.
To do this, you need to construct a perpendicular bisector between the two original points.
What fact can be stated about all points along the path of an angle bisector?
All points along an angle bisector are equidistant from the two lines that form the angle.