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Chapters 1.10 and 2.10 Measurement and Constructions

Part I: Measurement and constructions using a ruler, a set square and a protractor:

In this part, we will use a ruler, a set square and a protractor to:

Measure and draw straight lines to the nearest millimeter;

Measure and draw acute, obtuse and reflex angles to the nearest degree;

Construct a triangle in cases SAS or ASA (or equivalent variations of ASA) ;

Construct squares, rectangles and other regular polygons, given a side and an internal

angle;

Estimate the size of acute and obtuse angles;

A) Measure and draw straight lines:

1. Using a ruler, measure each of the lines below. Be as accurate as you can:

a)

___________________________________________________

b)

________________________

c) ________________

2. Using a ruler, draw lines of the following lengths:

a) 2.5 cm

b) 40 mm

c) 4.7 cm

d) 7.4 cm

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B ) Measuring and constructing angles:

Consider a line L . A point O L separates this line into two parts, each of which is called a

half – line (or a ray). O is called the origin of each ray.

An angle is the union of two half lines with the same origin O, as we can see in Figure 1 below:

Figure 1: An angle AOB

Angles are measured in degrees.

One degree is 1

360 of a full rotation (or, in other words, a full rotation has 360o ).

Example 1:

a) Using a protractor, measure this angle:

Make sure that the horizontal line aligns with the horizontal line of your protractor (line of

origin – of size 0o )

b) Draw an angle of size 120o

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Example 2:

For each of the following angles: Measure the angles. Find relationships between these

angles .

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C) Construct a triangle in cases SAS or ASA :

I. Case ASA :

Example: Draw a triangle ABC, where the length of the side AB is 8.2 cm,

52 and 40o oA B . Use only a ruler and a protractor.

To do this:

First, using a ruler, draw a segment of length 8.2 cm, and label it AB ;

Then, at the end A of your segment (perhaps the left end):

using a protractor, measure an angle of size 52o , and draw a line to form this angle ;

Similarly, at end B, using the protractor, measure an angle of size 40o , and draw a line to

form this angle .

The intersection of these two lines represent the vertex C of the given triangle.

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Extra practice: (Exercise 10.2 A Book 1):

Using the procedure outlined above, draw the following triangles ABC. After each construction, measure

and indicate the sides AB and BC, and the angle C.

1. AB = 7 cm ; 50 and 40o oA B

2. AB = 8.5 cm ; 30 and 60o oA B

3. AB = 8.3 cm ; 100 and 30o oA B

II. Case SAS:

Example: Given the sides AB = 8 cm, AC=7 cm and the angle 60oA , construct a triangle

using a ruler and a protractor.

To do this:

Draw the segment AB (base) of length 8 cm ;

Use your protractor and draw a line at the end A of your segment, which will form and

angle of 60o . Along this new line, measure 7 cm and label its endpoint C.

Draw a straight line from B to C to complete the triangle.

Extra practice: (Exercise 10.2 B Book 1):

Using the procedure outlined above, construct the following triangles ABC. For each triangle, measure

and indicate the other two angles : B and C.

1. AB =10 cm, AC =8 cm , 45oA

2. AB =8 cm, AC =10 cm , 100oA

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D. Construction of other geometrical figures:

Constructing a polygon accurately is similar with constructing a triangle, but it requires a very

careful use of a ruler and a protractor.

Example:

Construct a regular polygon of side length 4 cm and internal angles of size 108o .

How many sides does the obtained polygon have ?

Extra credit : Can you obtain the number of sides without using any drawing (e.g.

algebraically)?

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Extra practice: (Exercise 10.3 A and 10.3 B Book 1):

1. Construct the following shapes using a ruler and a protractor, or a ruler and a set

square:

2. a) Construct a regular polygon with side length 5 cm and internal angles of size 135o .

Name the polygon you have obtained.

b) Could you find the number of sides of the polygon without using any construction?

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3. Using a ruler, a protractor and a set square, construct the following shapes:

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Part II: Measurement and constructions using a ruler and a compass:

In this part, we use a ruler , a straight edge and a compass to:

Construct circles and arcs;

Construct a triangle in the case SSS ;

Construct a right triangle in the case HS ;

Draw the perpendicular bisector of a line segment;

Draw the bisector of an angle;

A. Constructing circles:

Definition: Given a point in plane O and a positive number r , a circle is the set of points in

plane which are at the fixed distance r from O.

Circles can be drawn using a pair of compasses. Make sure that the compass screw is tightened,

to prevent the compass from widening as the circle is drawn.

Example 1:

1. Draw the following circles using a pair of compasses. In each case, O is the center of the

circle.

B. Constructing triangles in the case SSS:

Example: Construct the triangle ABC with AB =10 cm, AC =8 cm and BC = 6 cm .

Steps:

Draw a segment AB of length 10 cm (the base of the triangle) ;

Open your compass to a length of 8 cm. Placing the compass at A, draw an arc (towards

the point B) using this opening of 8 cm of your compass.

Open your compass now to a length of 6 cm. Place the compass at B, and draw an arc

(towards the point A) using this new opening of the compass. Make sure that the two

arcs intersect;

The intersection of the two arcs is the point C, which generates the triangle ABC .

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Extra practice: (Exercise 10.2 A Book 2) :

For the following questions, construct the triangles PQR using only a ruler and a pair of

compasses. For each triangle, measure and indicate each of the angles P, Q and R.

1. PQ = 6 cm , PR = 8cm, QR = 10 cm

2. PQ = 4 cm, PR = 7 cm , QR = 7 cm

3. PQ = 6.2 cm, PR = 6.2 cm , QR = 6.2 cm

C. Constructing right angled triangles in the case hypotenuse side (HS):

A right angled triangle is a triangle with one angle of 90o . The side opposed to the 90o angle is

called the hypotenuse. The other two sides of the right angled triangle are called sides.

Theorem:

In a right angled triangle, the hypotenuse is longer than any of the other two sides.

Note: in the textbook a compass is used for drawing the right angle. We will use a set square

instead.

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Practice: (Exercise 10.2 B Book 2):

Construct the following right angled triangles using a ruler and a set square. For each triangle,

measure accurately the length of the third side:

2. In the obtuse angled triangle PQR, PX and XR are perpendicular to each other.

The length XQ = cm, PQ = 10 cm and PR = 15 cm.

a) Construct triangle PQR using only a ruler and a set square ;

b) Measure the length of QR ;

c) Measure the size of the angle PQR .

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D. Perpendicular bisector of a line segment:

Consider a segment PQ, as shown below.

Definition: The perpendicular bisector of PQ is the line ST such that:

a) PQST

b) ST divides PQ into two equal sides (it is perpendicular at the midpoint of PQ) .

Example: Consider a given segment AB, say 5 AB cm .

To draw the perpendicular bisector of AB, follow the following steps:

Open a pair of compasses to more than half the distance AB , but less than AB (say

about 3 cm ) ;

Place the compass at A and draw arcs of this length above and below AB;

Maintain the length and repeat this procedure, placing the compass at B now, such that

these two pairs of arcs intersect;

Draw a line through the two points where these arcs intersect;

This line is the perpendicular bisector of AB . Why ?

Note: Is there a simpler way to draw the perpendicular bisector of a given segment?

Extra practice (Exercise 10.3 Book 2):

1.

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2.

For the diagram above:

E. Draw the bisector of an angle

Just as bisecting a segment means dividing it into equal halves, bisecting an angle means

dividing the angle into two halves as well.

To find the bisector of an angle accurately , we use a compass (although a protractor can be

used as well, if done carefully).

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Example:

Consider the angle A given below:

Steps to bisect the angle A using a compass:

Open the compass to a small length (say 1 cm)

Place the compass at vertex A and these small arcs of length 1 cm, one for each side of

the angle. Obtain the points of intersection of the arc with the sides: B and C.

Keeping the same opening of the compass (of 1 cm), place the compass at B and draw

one arc inside the angle

Repeat this procedure placing the compass at C, drawing another arc inside the angle,

which intersects the arc from the previous step at point D;

The line AD is the bisector of angle A.

Extra practice: (Exercise Set 10.4 Book 2):

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5.

a) Construct an equilateral triangle of side length 6 cm.

b)

i) bisect angle A;

ii) Extend the bisecting line until it intersects side BC. Label this point X .

iii) Describe the triangle ABC.