Shape 5: Constructions and Loci

Shape 5: Constructions and Loci

Constructions and Loci

Must / Should / Could
Understand that a construction requires the use of correct geometrical equipment including a ruler, pair of compasses and angle measurer or protractor
Construct a triangle accurately given; The length of two sides and the angle formed at the intersection of those two sides or;
The length of one side and the two angles at the end of that side / Construct a triangle accurately given the lengths of all three sides using a pair of compasses and ruler
Understand the term, ‘congruent’ and identify congruent shapes / Know that congruent shapes are formed after transformations involving rotations, reflections and enlargements / Construct tessellations using appropriate congruent shapes (such as regular hexagons)
Understand the term, ‘similar’ and identify similar shapes / Know that ‘similar’ shapes have corresponding angles of the same size and lengths in the same ratio / Construct scale factor enlargement of given shapes (such as triangles)
Understand that the area of a similar shape will be equivalent to n2 × the area of the original shape, where the original shape has been enlarged by scale factor, n
Know the terms, ‘perpendicular’ and ‘bisect’
Construct the perpendicular bisector of a line segment / Know that a rhombus can be constructed by constructing the perpendicular of a line segment / Construct the perpendicular to a line which intersects the line segment at a given point
Construct the bisector of an angle / Construct an angle of 60o / Know that an equilateral triangle can be constructed by constructing an angle of 60o
Understand the terms, ‘locus’ and ‘loci’ / Construct the locus of all points equidistant from a point or a line segment / Construct the locus of all points equidistant from two given points
Key Words: construct, sketch, ruler, protractor, angle measurer, pair of compasses, vertex, vertices, side, line segment, congruent, similar, enlarge, scale factor, perpendicular, bisect / bisector, locus, loci, equidistant
Starters:
Sorting activity with shapes that have been constructed and shapes that have been sketched or drawn
Identify congruent shapes
Identify similar shapes
Practise measuring and drawing line segments and angles
What shape is this? How do we know? eg recapping properties of shapes (such as rhombus or kite)
Activities:
Construct congruent / similar shapes
Loci work: eg goat problems as in 10 ticks worksheets
Construct a diagram involving bearings / scale drawings
Plenaries:
Learning framework questions:
-What is the difference between a construction and a sketch or drawing?
-Following an enlargement, how many times does the original shape fit inside the enlargement? Is this what you expected?
-What does ‘bisect’ actually mean?
-What equipment would you expect to use for a construction?
Resources:
Pair of compasses / angle measurer / protractor / ruler
10 ticks worksheets
Possible Homeworks:
Investigate the golden ratio and shapes / patterns which follow the ratio
By re-examining the properties of shapes, construct common triangles, quadrilaterals and other polygons
Try to construct a 45 degree / 30 degree / 15 degree angle
Investigate the pattern for the number of parts that are formed by bisecting a line segment, then bisecting each bisected part and so on.
Describe the locus for an everyday moving object. Does it follow a predictable pattern or is it random / chaotic?
Teaching Methods/Points:
CONSTRUCTION
Students should know that most constructions require the use of a pencil, a ruler, a pair of compasses and an angle measurer or protractor. A sketch or drawing will still require the use of a pencil and ruler in Maths, but the important difference between a sketch and a construction is the accuracy of the lengths and angles in the diagram. Construction lines should remain clear, the geometric equivalent of ‘showing your workings / process’.
The topic of construction, in particular, provides an opportunity to remind students of some important 2D shapes and of their important properties. For example;
A rhombus has diagonals which perpendicularly bisect each other.
A kite has just one diagonal which perpendicularly bisects the other.
An equilateral triangle has three angles of 60o.
It is also important to ensure students are familiar and conversant with the following terms;
Perpendicular lines are two lines that meet each other at right angles (90 degrees)
Bisect means to divide or cut something into 2 equal parts, so a bisector is a line which cuts another line into 2 equal parts.
Construct means use appropriate geometric equipment to make an accurate drawing.
Locus (plural loci) is the pathway of a moving object
CONSTRUCTING A TRIANGLE
The simplest and most familiar task in terms of using prior skills is that of constructing a triangle requiring precise measurement of length and measurement of angle. This is covered as part of ‘Shape 1’ as a means to consolidate skills drawing line segments and angles. In order to develop construction skills, tasks can be structured as follows;
1) To construct a triangle given the length of line segment, AB, the size of angle BAC and the length of AC;
Required: Ruler, angle measurer or protractor
Ensure students are required to measure and draw line segments in both centimetres (eg 3.4 centimetres) and millimetres (eg 34 millimetres) and stretch them to make judgements on even more precise measurements such as (34.5 millimetres). The accuracy of the angle is important and this provides an opportunity to practise using angle measurers and protractors.
2) To construct a triangle given the length of line segment, AB and the size of angles BAC and ABC;
Required: Ruler, angle measurer or protractor
Students should be encouraged to draw line segments AC and BC faintly at first (and beyond the length that they are likely to need in order to ensure the line segments intersect) once line segment AB has been accurately drawn and the angles have been measured.
3) To construct a triangle given the length of three line segments, AB, BC, and AC.
Required: Ruler, pair of compasses
This construction relies on the understanding that the intersection of the arcs drawn from centre points A and B respectively (once line segment AB has been drawn accurately), with radii equal to the lengths BC and AC respectively, provides the location of point C. They must avoid the temptation to simply estimate the location of point C by using a ruler and a trial and error approach!

OTHER CONSTRUCTIONS
Constructing a perpendicular bisector of a line
Draw line segment AB.
Set the radius of the pair of compasses to be more than half of length AB.
Put the point on A and draw an arc above and below the line.
Put the point on B and draw an arc above and below the line.
Make sure the two arcs cross each other.
Label the points where the two arcs cross, C and D then draw line segment CD.
CD is the perpendicular bisector of AB.






Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P
Draw line segment AB.
Identify point P which is along the length of line segment AB.
Measure and mark an equal distance away from point P in either direction along line segment AB.
Ensure the radius of the pair of compasses is sufficient to allow for the arcs to intersect.
Put the point on the marks (placed an equal distance away from point P) and draw arcs above and below the line.
Make sure the two arcs cross each other.
Label the points where the two arcs cross, M and N then draw line segment MN.
MN is a line perpendicular to AB which meets AB at a specific point, P.






Constructing a bisector of an angle
Draw angle ABC.
Measure and mark an equal distance along line segments BA and BC from corner point B.
Set the radius of the pair of compasses to ensure that the arcs will intersect.
Draw arcs from the marked points (an equal distance away from point B) so that the arcs intersect.
Draw a line which passes through B and D, the point of intersection of the two arcs.
BD is the bisector of angle ABC.
A
D
B


C
Constructing a 60 degree angle
Draw line segment AB.
Set the radius of the pair of compasses to exactly the same length as the length of line segment AB.
Draw an arc from point A.
Draw an arc from point B, such that it intersects the first arc at point C
Draw line segment AB.



SIMILARITY AND CONGRUENCE
Congruent shapes have exactly the same size and shape (as explained in Shape 3: Transformations). Students must understand that shapes that have been translated, reflected and rotated are always congruent.
Similar shapes have corresponding angles of the same size, and corresponding lengths in the same ratio. Any scale factor enlargement of a shape creates a similar shape.
For example;

scale factor 2 enlargement
Here are examples of other shapes that are similar (with corresponding angles of the same size and corresponding sides in the same ratio.)
These examples are also given in the notes for enlargement (Shape 4: Transformations)

eg



Shapes A and Bare similar because the corresponding angles are the same size and all of the lengths of Shape B are exactly twice the size of the lengths of Shape A. The ratio of the lengths is 1 : 2
Shapes C and Dare similar because the corresponding angles are the same size and all of the lengths of Shape D are three times the size of the lengths of Shape C. The ratio of the lengths is 1 : 3
Shapes E and Fare similar because the corresponding angles are the same size and all of the lengths of Shape F are 2.5 times the size of the lengths of Shape E. The ratio of the lengths is 1 : 2.5 [this can be written as 2 : 5]
Every length is twice as long. It is useful to investigate the resulting increase in area, leading to the conclusion that for a scale factor enlargement of n, the area will be n2 times as big. Similarly, applying the same investigative approach to volume will lead to the conclusions that a scale factor enlargement of n for a solid will lead to a volume which is n3 times as big.
Incorporate activities which require students to identify similar shapes (by considering the ratio of their lengths.
While work on constructions is not directly linked to ‘congruent’ and ‘similar’ shapes (which are actually dealt with as part of the topic of ‘Transformations’), this is an opportunity to consolidate student understanding and extend the nature of the tasks. For example, given a triangle ABC, such that AB = 3cm, BC = 4 cm and angle ABC is 60 degrees, construct the enlargement of triangle ABC by scale factor 2.
LOCI
A locus (or loci in the plural) is defined as the path of a moving point. Students must understand the term;
equidistant meaning “equal distance away”
You have to be able to draw a locus given a description of the rule that the locus follows, and you have to be able to describe the rule that a locus follows.

Locus of points a fixed distance from point A

Locus of points a fixed distance from line AB





Construction for locus of points equidistant
from points A and B



B
Construction of locus of points equidistant
from AB and AC



C
The topic of loci provides an opportunity to apply construction skills to a range of real-life problems. While the above examples demonstrate specific skills that students should learn when drawing a locus, and illustrate the type of descriptions that students need to give in words given a diagram of a locus, the topic can be used to deal with other real-life problems typified by a goat in a field. These problems generally require students to use their geometric equipment to construct specific scenarios, such as a goat tethered at a particular point with a rope of a certain length. These problems are important in developing geometric reasoning skills!
Constructions: Help Sheet
Construct means use appropriate geometric equipment to make an accurate drawing.
Perpendicular lines are two lines that meet each other at right angles (90 degrees)
Bisect means to divide or cut something into 2 equal parts
Constructing a triangle given the length of three line segments, AB, BC, and AC.


Constructing a perpendicular bisector of a line






Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P







Constructing a bisector of an angle
A

D
B

C
Constructing a 60 degree angle




Loci: Help Sheet
A locus (or loci in the plural) is the pathway of a moving point.
Equidistant meaning “equal distance away”
Examples of common loci:

Locus of points a fixed distance from point A

Locus of points a fixed distance from line AB





Construction for locus of points equidistant
from points A and B



B
Construction of locus of points equidistant
from AB and AC



C
LOCI AND GRAPHS (equation of a circle)

Using a coordinate grid, you may be asked to
construct the locus of a circle given an ‘equation’,
or describe the locus of a circle by writing the
equation.
The equation of a circle is given by the equation
x2 + y2 = r2

So, for example; x2 + y2 = 25 is a circle with centre
on the origin, r2 = 25, so radius is 5. The locus
is shown here

Shape 5: Constructions and Loci