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Angle rules Angles around a point Angles in a triangle Angles in parallel lines TrigonometryThis topic is relevant for:
Here we will learn about loci, constructions and bearings, including drawing various constructions, using constructions when drawing loci and measuring and calculating bearings.
There are also loci & construction worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.
Loci and construction are geometric objects that are drawn with specific features.
To draw loci and constructions accurately, you will be required to use a ruler, a pair of compasses, and a protractor when appropriate.
Constructions are set methods for creating accurate drawings in maths.
A locus of points (plural loci) is a set of points that follow a given rule. We often use constructions to accurately draw loci.
A bearing is an angle, measured clockwise from north, used to describe the position of an object. Bearings are given using three figures, for example katex is not defined
There are a number of different constructions that you need to be able to perform for GCSE maths. These include,
To do constructions we carefully follow the method, leaving all construction lines visible.
The most common constructions are,
When doing constructions, all construction lines should be left visible.
Step-by-step guide: Constructions
A locus of points (plural loci) is a set of points that follow a given rule. Given a rule, we need to be able to identify the points that follow that rule.
Here are some examples,
Step-by-step guide: Loci
A bearing is an angle, measured clockwise from north, used to describe the position of an object. They are written using three figures. Bearings can be drawn, measured or calculated depending on the problem.
The bearing of B from A is katex is not defined This means to get from A to B, you need to travel in a direction katex is not defined clockwise from north.
We can find the bearing of A from B using angle rules.
As both north lines in the diagram below are in the same direction, these lines are parallel. This means that the angle at A and the angle at B are co-interior (their angle sum is katex is not defined).
The angle katex is not defined is equal to katex is not defined
The back bearing is the bearing of A from B. As the sum of angles at a point is katex is not defined the bearing katex is not defined of A from B in the diagram below is katex is not defined
Step-by-step guide: Bearings
Get your free loci and construction worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free loci and construction worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEBisect the angle ABC.
2With the point of the compasses on each intersection, draw two further arcs.
3Draw a line from the vertex to the intersection of the arcs.
Step-by-step guide: Angle bisector
Draw the locus of points equidistant from lighthouse katex is not defined and lighthouse katex is not defined.
The locus of points equidistant from two points is the perpendicular bisector of the line between those points. Imagine a straight line connecting the two points katex is not defined and katex is not defined.
We need to construct the perpendicular bisector of the line katex is not defined.
With the point of the compasses on one end of the line, draw an arc.
Here we will place the point of the compass at katex is not defined and draw an arc.
The radius of the arc must be greater than half of the distance between katex is not defined and katex is not defined.
With the point of the compasses on the other end of the line, draw a second arc.
Placing the point of the compass at katex is not defined, we construct another arc with the same radius as the previous arc.
The radius of each arc must be the same, otherwise the two points of intersection will not be the same distance from katex is not defined and katex is not defined.
Draw a straight line through the two points where the arcs cross.
Using a ruler, draw a straight line through both pairs of intersecting arcs.
The line we have drawn is the locus of points equidistant from katex is not defined and katex is not defined.
Step-by-step guide: Perpendicular bisector
Two phone masts, katex is not defined and katex is not defined are katex is not defined apart. The signal from phone mast katex is not defined can reach katex is not defined and the signal from phone mast katex is not defined can reach katex is not defined Using the scale katex is not defined shade the area covered by signal from both phone masts.
Use the wording of the region required to decide what constructions are needed.
In this case, we need to draw a circle with radius katex is not defined (to represent katex is not defined) around mast katex is not defined and a circle with radius katex is not defined (to represent katex is not defined) around mast katex is not defined.
Perform any relevant constructions for points or line segments involved.
For the first circle with centre katex is not defined, measure the distance between the point of the compass and the pencil to katex is not defined using a ruler. Place the point of the compasses on katex is not defined and draw a circle. The radius of this circle is katex is not defined
For the second circle with centre katex is not defined, measure the distance between the point of the compasses and the pencil to katex is not defined using a ruler. Place the point of the compasses on katex is not defined and draw a circle. The radius of this circle is katex is not defined
Indicate the region required as necessary.
The shaded area is covered by signals from both phone masts.
Step-by-step guide: Constructions between points and lines
Below is a diagram of Ellieβs garden. Ellie wants to plant a new tree. She wants the tree to be closer to the side katex is not defined than the side katex is not defined, more than katex is not defined from katex is not defined and less than katex is not defined from the side katex is not defined. Using the scale katex is not defined indicate the area in which Ellie can plant the tree.
Use the wording of the region required to decide what constructions are needed.
Breaking down the wording of the question we have the following three facts,
To determine the points closer to the side katex is not defined than the side katex is not defined, we need to construct an angle bisector. This is because all of the points that are on the angle bisector are the same distance from both lines, and so points in one half will be closer to one line than the other.
To determine the points more than katex is not defined from katex is not defined, we need to construct a circle with radius katex is not defined This is because katex is not defined is a point and we need all the points beyond katex is not defined from katex is not defined.
To determine the points less than katex is not defined from the side katex is not defined, we need to construct a parallel line that is katex is not defined from the line katex is not defined.
Perform any relevant constructions for points or line segments involved.
Indicate the region required as necessary.
The shaded area is the locus of points which satisfy all of Ellieβs criteria,
The bearing of a ship from lighthouse katex is not defined is katex is not defined The bearing of the same ship from lighthouse katex is not defined is katex is not defined Mark the position of the ship on the diagram.
Locate the point you are measuring the bearing from and draw a north line if there is not already one given.
We need to draw bearings from both points katex is not defined and katex is not defined so both need a north line.
Using your protractor, place the zero of the scale on the north line and measure the required angle clockwise, make a mark on your page at the angle needed.
Draw a line from the start point in the direction of the bearing. If you are producing a scale drawing and know the distance to locate a point use this scale appropriately.
The ship is located at the point where the two lines intersect.
A plane starts at airport katex is not defined. The plane flies katex is not defined east and then katex is not defined north before landing at airport katex is not defined. Find the bearing of airport katex is not defined from airport katex is not defined.
Locate the points you are calculating the bearing from and to.
Drawing a diagram is very helpful here.
Using the north lines for reference at both points, use angle rules and/or trigonometry to calculate any angles that are required.
We can find angle katex is not defined using SOHCAHTOA and then use angle facts to determine angle katex is not defined
katex is not defined
As the sum of angles in a right angle is katex is not defined degrees, we can find the missing angle katex is not defined by subtracting katex is not defined from katex is not defined to get katex is not defined
Read off the three-figure bearing required.
Converting the angle of katex is not defined to a three-figure bearing gives us the solution katex is not defined
Step-by-step guide: Constructing triangles
See also: How to draw a hexagon
When drawing constructions, the construction lines must be shown and not erased.
When drawing the locus around a polygon or a line, the locus around a vertex should be an arc. For example,
When finding the locus of points equidistant from two sides of a rectangle, we need to bisect the angle rather than drawing a line between the two corners.
For example,
It is important to remember that bearings are always measured from north.
Bearings must be measured in a clockwise direction.
For example, the bearing of katex is not defined from katex is not defined is katex is not defined not katex is not defined
1. What type of construction is shown below?
Angle bisector of the angle katex is not defined
Perpendicular to the line katex is not defined at the point katex is not defined
Perpendicular bisector of the line katex is not defined
Triangle katex is not defined
The two straight lines are at right angles to each other and so they are perpendicular lines. Also, the radius of each arc from katex is not defined and katex is not defined is the same and so the line is bisected into two equal halves.
2. Which diagram correctly shows how to construct the angle bisector of angle katex is not defined?
An arc of the same radius is drawn on line katex is not defined and katex is not defined from the point katex is not defined. From these arcs, two intersecting arcs are drawn with a straight line connecting the vertex at katex is not defined and the intersection point.
3. Which diagram shows the locus of points a distance of katex is not defined from the triangle?
Each vertex should be treated as a point and therefore the locus around each vertex should be an arc. The locus of points from a line are parallel.
4. Hannah wants to shade the area closer to side katex is not defined than side katex is not defined, and more than katex is not defined from point katex is not defined. Which diagram shows the correct area?
We need to draw a circle with radius katex is not defined around the point katex is not defined and the angle bisector of angle katex is not defined. The shaded region is outside of the circle and to the left side of the line drawn for the angle bisector.
5. Find the bearing of katex is not defined from katex is not defined.
Bearings are measured clockwise from north. The bearing of katex is not defined from katex is not defined is katex is not defined
6. The bearing of airport katex is not defined from airport katex is not defined is katex is not defined What is the bearing of airport katex is not defined from airport katex is not defined?
By drawing a diagram, we can see that the co-interior angle at katex is not defined is katex is not defined (remember that all north lines are parallel). Therefore the bearing of katex is not defined from katex is not defined, measured clockwise from north, is katex is not defined
1. Construct the perpendicular to the line katex is not defined at the point katex is not defined.
(3 marks)
Two arcs equidistant from katex is not defined.
(1)
Two further arcs above (or below) katex is not defined.
(1)
Perpendicular drawn and all construction lines shown.
(1)
2. Shade the area that is closer to the side katex is not defined than the side katex is not defined and is closer to the point katex is not defined than the point katex is not defined.
(3 marks)
Perpendicular bisector of katex is not defined.
(1)
Angle bisector of angle katex is not defined.
(1)
Correct area shaded.
(1)
3. Some rangers are tracking the position of a lion.
The lion is on a bearing of katex is not defined from ranger katex is not defined and katex is not defined from ranger katex is not defined.
Mark the position of the lion on the diagram below.
(3 marks)
Bearing of katex is not defined from katex is not defined drawn.
(1)
Bearing of katex is not defined from katex is not defined drawn.
(1)
Position of lion marked.
(1)
You have now learned how to:
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