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Construct different types of triangles

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 8

RATIO AND PROPORTION

MEASUREMENT

CONSTRUCTIONS

Module 17

CONSTRUCTING DIFFERENT ANGLES AND TRIANGLES

ACTIVITY 1

Constructing different angles and triangles

[LO 3.4, 3.5, 4.7]

1. Drawing an angle:Requirements: pencil, ruler, protractor.

1.1 Always begin by drawing a base line.

1.2 Make a mark, e.g. on the left, and position the protractor on the mark.

1.3 Read your protractor from 0°.

1.4 In the case of an angle that is larger than 180°, the relevant angle size must be deducted from 360° before it is drawn. The angle around the outside (the reflex angle) is the angle that you will have to draw.

E.g. 320°: (360° – 320° = 40°). Draw a 40°angle. The reflex angle now represents the 320°.

2. Construct the following angles and name each one:

  • ABˆBˆ size 12{ { hat {B}}} {}C = 75°

Type of angle:

2.2 CDˆDˆ size 12{ { hat {D}}} {}E = 135°

Type of angle:

2.3 FGˆGˆ size 12{ { hat {G}}} {}H = 215°

Type of angle:

3. Constructing a triangle:

Requirements: pencil, ruler, protractor and pair of compasses.

  • Always begin by making a rough sketch.
  • Then use one of the sides of which the length is provided as a base.
  • E.g. construct ΔΔ size 12{Δ} {}ABC with BC = 40 mm, BˆBˆ size 12{ { hat {B}}} {}= 70° and CˆCˆ size 12{ { hat {C}}} {}= 50°.

Rough sketch:

Figure 1
Figure 1 (Picture 200.png)
  • To measure a lateral length accurately, you must measure the length on you ruler with the help of a pair of compasses. Then the compass point must be positioned on B and the position of C must be indicated with a pencil mark.
  • Construction:

4. Construct each of the following triangles:

4.2 ΔΔ size 12{Δ} {}PQR with QR = 58 mm, PQˆQˆ size 12{ { hat {Q}}} {}R = 62° and QPˆPˆ size 12{ { hat {P}}} {}R = 69°.

Measure:

  1. a) PQ = mm
  2. b) RˆRˆ size 12{ { hat {R}}} {} =

4.2 Isosceles ΔΔ size 12{Δ} {}ABC with BC = 42 mm, AB = AC and BˆBˆ size 12{ { hat {B}}} {} = 63°.

Measure:

a) PQ = mm

ACTIVITY 2

Bisecting any given line or angle

[LO 3.4, 3.5, 4.7]

  1. Bisecting a given line AB:
Figure 2
Figure 2 (Picture 201.png)
  • Measuring line segment AB (e.g. 40 mm).
  • Using a pair of compasses, measure slightly more than half of the line(i.e. ± 22-25 mm).
  • Position the point of the pair of compasses on A and make a pencil stroke below and above the line.
  • Position the point of the compasses on B and draw another pencil stroke above and below the line.
  • Connect the intersections of the pencil strokes.
  • Name the point on line AB, P. P is the centre of line AB.

2. Now try the following:

  • Draw line segment PQ = 70 mm.
  • Bisecting line segment PQ, as in nr. 1 explained.

Figure 3
Figure 3 (Picture 13.png)

3. Bisect πABC:

  • Place the point of the pair of compasses on B.
  • Draw an arc of any size as indicated.
  • Position the point of the compass on the point where the two lines intersect and draw pencil lines inside the angle.
  • Position the point of the compass on the other point of intersection and draw a line inside the angle, so that the two lines intersect.
  • Connect BˆBˆ size 12{ { hat {B}}} {} (angle B) with the point where your pencil lines intersect.
  • BˆBˆ size 12{ { hat {B}}} {}1 will have the same size as BˆBˆ size 12{ { hat {B}}} {}2. Measure both angles. Are they equal?

4. Try the following:

  • Draw DEˆEˆ size 12{ { hat {E}}} {}F = 125°.
  • Bisect DEˆEˆ size 12{ { hat {E}}} {}F.

ACTIVITY 3

To construct a line perpendicular from a given point to another line

[LO 3.4, 3.5, 4.7]

1. Construct AD size 12{ ortho } {}BC.

  • Place your compass point on A (you want to draw a perpendicular line on BC from A.)
  • Make an arc on BC.
  • Place the point of your compasses on the one point of intersection between the arc and BC. Draw a line below BC. Place the point of your compasses on the other point of intersection between the arc and BC and draw another line below BC, so that the two lines intersect.
  • Connect A with the intersection of the two drawn lines.
  • Mark the point of intersection D.
  • AD will be perpendicular to BC. (AD size 12{ ortho } {}BC.)

Figure 4
Figure 4 (Picture 21.png)

2. Try doing the following:

  • Draw any acute-angled ΔΔ size 12{Δ} {}PQR.
  • Construct PS size 12{ ortho } {}QR.
  • What is the meaning of PS size 12{ ortho } {}QR?

ACTIVITY 4

Constructing inscribed and circumscribed circles

[LO 3.4, 3.5, 4.7]

1. Constructing a circumscribed circle:

  • Draw any acute-angled triangle.
  • Bisect all three angles. You will notice that the tree bisecting lines meet in a single point.
  • Try to locate the distance where you could position your compass to draw a circle within or around the triangle.
  • Explain what the distance was at which you were able to draw an accurate circle around the triangle.

  • What is this distance called?

  • What type of circle could you draw?

1.7 Conclusion: A . circle can be constructed by

bisecting the of a triangle.

2. Constructing an inscribed circle:

  • Draw any acute-angled triangle.
  • Bisect all three angles. You will notice that the tree bisecting lines meet in a single point.
  • Try to locate the distance where you could position your compass to draw a circle within or around the triangle.
  • Explain what the distance was at which you were able to draw an accurate circle inside the triangle.

  • What is this distance called?

  • What type of circle could you draw?

2.7 Conclusion: A circle can be constructed by

bisecting the of a triangle.

ACTIVITY 5

Constructing a line parallel (ll) to a requested line with the help of a pair of compasses

[LO 3.4, 3.5, 4.7]

1. Required: construct FA ll QR, so that AR = 30 mm.

1.1 Draw an imaginary line (dotted line) FA where the parallel line is required to be.

1.2 Mark A on PR so that AR = 30 mm.

1.3 Position the point of your compasses on R and draw an arc (any size) as indicated.

1.4 Maintaining the setting of your pair of compasses (same size), place the point on A and draw an arc like the previous one.

1.5 Measure the distance, marking it with crosses (x) as indicated.

1.6 Place the compass point on the circle (o) as indicated. This line will intersect the arc and should be on the imaginary line.

1.7 Connect A with the intersecting point of the last drawn line.

1.8 Mark F on PQ. FA will be parallel to QR.

1.9 What does it mean when we say that FA ll QR?

Figure 5
Figure 5 (Picture 202.png)

2. Try doing the following by yourself:

  • Construct any obtuse-angled ΔΔ size 12{Δ} {}PQR.
  • Bisect PR and designate the centre F.
  • Draw a line through F parallel to QR.
  • The parallel line PQ must intersect G.

ACTIVITY 6

Constructing a parallelogram

[LO 3.4, 3.5, 4.7]

1. You are the owner of a farm in Mpumalanga. You wish to reward one of your farm workers, Michael Mohapi, for his good service of the past 20 years. You present Michael with a stretch of land as a gift. The precondition is that the land must be measured out in the form of a parallelogram according to measurements indicated on a plan.

1.1 The first problem that arises has to do with the fact that Michael does not know what a parallelogram is. Use a sketch to provide Michael with all the characteristics of a parallelogram.

1.2 Also show Michael the mathematical “abbreviation” for a parallelogram, so that he will know what is meant when he sees the relevant "sign".

1.3 Now you have to execute a construction to indicate exactly how the land is to be measured.

Assessment

Table 1
LO 3
Space and Form (geometry)The learner is able to describe and represent features of and relationships between two-dimensional forms and three-dimensional objects in a variety of orientations and positions.
We know this when the learner:
3.2 describes and classifies geometric figures and three-dimensional objects in terms of properties in contexts inclusive of those that can be used to promote awareness of social, cultural and environmental issues, including:3.2.1 sides, angles and diagonals and their relationships, focusing on triangles and quadrilaterals (e.g. types of triangles and quadrilaterals);
3.3 uses vocabulary to describe parallel lines that are cut by a transverse, perpendicular or intersection line, as well as triangles, with reference to angular relationships (e.g. vertically opposite, corresponding);3.4 uses a pair of compasses, a ruler and a protractor for accurately constructing geometric figures so that specific properties may be investigated and nets may be designed;3.5 designs and uses nets to make models of geometric three- dimensional objects that have been studied in the preceding grades and up till now;3.7 uses proportion to describe the effect of expansion and reduction on the properties of geometric figures;3.8 draws and interprets sketches of geometric three-dimensional objects from several perspectives, focusing on the retention of properties.
LO 4
MeasuringThe learner is able to use appropriate measuring units, instruments and formulas in a variety of contexts.
We know this when the learner:
4.1 solves more complicated problems involving time, inclusive of the ratio between time, distance and speed;4.2 solves problems involving the following:4.2.1 length;4.2.2 circumference and area of polygons and circles;4.2.3 volume and exterior area of rectangular prisms and cylinders;
4.3 solves problems using a variety of strategies, including:4.3.1 estimation;4.3.2 calculation to at least two decimal points;4.3.3 use and converting between appropriate S.I. units;
4.5 calculates the following with the use of appropriate formulas:4.5.1 circumference of polygons and circles;4.5.2 area of triangles, right angles and polygons by means of splitting up to triangles and right angles;4.5.3 volume of prisms with triangular and rectangular bases and cylinders;
4.7 estimates, compares, measures and draws triangles accurately to within one degree.

Memorandum

ACTIVITY 1 – ACTIVITY 5

The memorandum of this learning unit is done by the learners and /or determined by the teacher for corrections.

ACTIVITY 6

1. Both pairs opposite sides are equal.

2. Both pairs opposite sides are parallel.

3. Both pairs opposite angles are equal.

4. Diagonals bisect each other.

5. One pair opposite sides – equal and parallel.

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