Mathematics

Key unit competence

By the end of this unit, the learner should be able to solve shape problems about
enlargement and similarities in 2D.

Unit Outline

• Definition and properties of similarity.

• Similar polygons

• Similar triangles

• Finding length of sides of shapes using similarity and Thales theorem

• Definition of enlargement

• Properties of similarity

• Determining linear scale factor of enlargement

• Determining the centre of enlargement

• Enlargement in the Cartesian plane

• Areas and volumes of similar shapes and objects respectively

• Composite and inverse enlargement

Introduction

Unit Focus Activity

One of the coffee processing and marketing companies in Rwanda
packages its coffee sachets in small cartons for domestic use and export.
Suppose that each carton measures24 cm long, 12 cm wide and 16 cm high.
The company is planning to make a very big model of the carton, 1.8 m long to
be mounted on the roadside of a busy highway for advertisement.
The company manager was told that

you learn Mathematics in school. So, he has invited you to advice them on
all the dimensions (measurements) they should use in making the model, so that
it will have exactly the same shape as the carton.

1. Identify and state two concepts in Mathematics that will help you
    determine the correct dimensions of the model.

2. Use these concepts to determine the correct dimensions of the
     model.

3. Compare your dimensions with those obtained by other classmates
     in class discussion.

4. How can you advice the company on the disposal of used-up cartons.
    Drink Rwanda grown coffee and use other Rwanda made products to grow
    our economy and support our farmers and producers

11.1 Similarity

11.1.1 Similar triangles

Activity 11.1
1. Use a mathematics dictionary, reference books or Internet
    to find out the definition of similarity

2. Draw a triangle with sides 4.6 cm, 3.0 cm and 3.4 cm,
    measure its angles.

3. Draw another triangle PQR whose lengths are 1 12
   times as long as the first triangle of ABC.

4. Measure the angles of the two triangles. What do you notice?

5. Compare the shapes of the two triangles. What do you notice.
   Explain your observation.

Activity 11.2

Activity 11.3

2. Measure the remaining one angle in each triangle and compare.

3. Prove whether ΔABC is similar to ΔDEF.

Activity 11.4

Let us consider two similar triangles DABC and DDEF. (Fig 11.6)

We say that triangles ABC and DEF are
similar.
Generally, two triangles are similar if:

1. If two pairs of corresponding angles are equal, then the other remaining
    pair of corresponding angles must also be equal.

2. If the ratios of two pairs of corresponding sides is the same and
    angles the between those considered sides in each triangle are equal,
    then the ratio will also be the same one for the remaining two pairs of
    corresponding sides.Similarity is denoted by the symbol “∼”.
    So, for the above two similar triangles,we write: ABC∼DDEF

Mathematically, two triangles are said
to be similar, if one of the following three criteria hold:

1. AAA or AA criterion: Two triangles are similar if either all the three
   corresponding angles are equal or any two corresponding angles are
   equal. AAA and AA criteria are same because if two corresponding
   angles of two triangles are equal, then third corresponding angle will
    definitely be equal.

2. SSS criterion: Two triangles are said to be similar, if all the corresponding
    sides are in the same proportion.

3. SAS criterion: Two triangles are similar if their two corresponding

sides are in the same proportion and the corresponding angles between
these sides are equal. Similar figures have the same shape
irrespective of the size.
Note: Would the statement still be true if ‘triangles’ is replaced with ‘polygons’? The
answer is ‘No’: With all polygons other than triangles, the ‘or’ must be replaced
with ‘and’.

Exercise 11.1

1. State if, and why, the pairs of shapes in Fig. 11.9 are similar.

2. Construct two triangles ABC and PQR with sides 3 by 3 by 5 cm and 12 by
    12 by 20 cm respectively. Measure all the angles. Are triangles ABC and
     PQR similar? Are all isosceles triangles similar? Are all equilateral triangles
       similar?

3. The vertices of three right-angled triangles are given below:
A(3 , 3), B(4 , 5), C(3 , 5);
P(1 , 3), Q(1 , 5), R(2 , 4);
X(–2 , 3), Y(1 , –1), Z(–2 , –1).
Which two triangles are similar?

4. In Fig. 11.10 ∠QFR = 90º and∠QER = 90º.

11.1.2 Similar polygons

Activity 11.5

1. (a) Draw a rectangle ABCDmeasuring 3 cm by 4 cm.

   (b) Draw another rectangle EFGH measuring 4 cm by 6 cm.

   (c) What do you notice about the  two rectangles? Find the ratio
       of the corresponding sides.

(d) What do you notice?

2. (a) Draw a rectangle PQRSmeasuring 2 cm by 4 cm.

(b) Draw another rectangle UVWX measuring 5 cm by 10 cm.

(c) Find the ratio of corresponding sides of the rectangles.

(d) What do you notice? Two or more polygons are similar if the
      ration of the corresponding sides is constant and the corresponding angle are equal.

Example 11.2

Consider rectangle WXYZ of length 2 cm and width 3 cm and rectangle
QRST of the lengths 3 cm width 5 cm. Find the ratio of their corresponding
sides.

Activity 11.6

Example 11.4
Consider the shapes in Fig 11.15. Which among these shapes are similar
to Fig 11.15 (a).

Solution

Fig 11.15 (d) is similar to Fig 11.15 (a).

The corresponding angles are equal and the ratio of the corresponding sides is
 constant.

Example 11.5

Determine whether the hexagons in Fig. 11.16 are similar. State your
reasons.

Remember that:
Two polygons are similar if:

1. The ratio of corresponding sides is constant

2. The corresponding angles are equal.

Exercise 11.2

1. State if, and why, the pairs of shapes in Fig. 11.17 are similar.

2. Measure the length and breadth of this text book. Measure also the length and
    breadth of the top of your desk. Are the two shapes similar? If not, make a scale
   drawing of a shape that is similar to the top of your desk.

3. A photograph which measures 27 cm by 15 cm is mounted on a piece of card so
    as to leave a border 2 cm wide all the way round. Is the shape of the card similar to
    that of the photograph? Give reason for your answer.

11.1.3 Calculating lengths of sides of similar shapes
           using similarity and Thales theorem

Activity 11.7

Consider the rectangles PQRS and WXYZ in Figure 11.18 below.

We can use this fact to calculate the lengths of sides of the triangles.
Remember that also, similar traingles and trapezia also obey Thales theorem which
states that:

“If a line is drawn parallel to one side of a triangle intersecting the other two sides,
it divides the two sides in the same ratio. For example, in triangle PQR below, ST
is parallel to PQ.

Example 11.6

Solution

Separating the triangles

Example 11.8

Fig. 11.26 and Fig. 11.31 below are similar. Find the lengths of sides marked
with letters.

Example 11.9

Fig. 11.28 shows two triangles ABC and PQR. Calculate the lengths BC and PQ.

Solution

Since the corresponding angles are equal, Δs ABC and PQR are similar.

∴ the ratio of corresponding sides is constant.

Example 11.10

In the figure below, DE is parallel to BC, find the value of x.

Exercise 11.3

1. The triangles in each pair in Fig. 11.31 are similar. Find x.

2. Show that the two triangles in Fig. 11.32 are similar. Hence calculate
    AC and PQ.

3. Find the values of x and y in Figure 11.33 (a) and (b).

4. Which two triangles in Fig. 11.34 are similar? State the reason. If
     AB = 6 cm, BC = 4 cm and DE = 9 cm, calculate BD.

5. Find the value of x from the figures 11.35 and 11.36 below:

6. In Fig. 11.37, identify two similar triangles. Use the similar triangles to
    calculate x and y.

7. A student used similar triangles to find the distance across a river. To construct

    the triangles she made the measurements shown in Fig. 11.38. Find the distance
    across the river.

11.1.4: Definition and properties of similar solids     

         Activity 11.8

3. What do you notice about the ratio of corresponding sides?

4. Measure the corresponding angles ∠A and ∠A′, ∠B and ∠B′,
    ∠C and ∠C′, etc. What do you notice?

Consider a small cuboid measuring 8 cm × 12 cm × 6 cm and big cuboid
measuring 12 cm × 18 cm × 9 cm shown in Fig 11.40.

Lets compare the ratios of corresponding sides

We notice that the ratio of corresponding
sides is a constant (same).
Lets compare the corresponding angles.
∠ABC = ∠PQR, ∠ABF = ∠PQV,

∠BCG = ∠QRV

We notice that the all corresponding angles are equal.
We say that cuboid ABCDEFGH is similar to cuboid PQRSTUVW.
Note that even naming similar figures we
match the corresponding sides e.g. AB
corresponds to PQ, CD coresponds to RS and so on.
Two solids are similar if:

1. The ratio of the lengths of their corresponding sides is constant.

2. The corresponding angles are equal.

Example 11.12

Determine whether the following pairs objects are similar or not.

Not all ratios of corresponding sides are equal i.e the ratio of corresponding sides
is not a constant. Hence, the two prisms are not similar.

Example 11.13

A jewel box, of length 30 cm, is similar to a matchbox. If the matchbox is 5 cm long,
3.5 cm wide and 1.5 cm high, find the breadth and height of the jewel box in Fig. 11.42.

Solution

The ratio of the lengths of the jewel box and the matchbox is

This means that each edge of the jewel box is 6 times the length of the
 corresponding edge of the matchbox. Width of jewel box = 6 × 3.5 cm = 21 cm,
 and height of jewel box = 6 × 1.5 cm = 9 cm.

Exercise 11.4

1. A scale model of a double-decker bus is 7.0 cm high and 15.4 cm long. If
   the bus is 4.2 m high, how long is it?

2. A cuboid has a height of 15 cm. It is similar to another cuboid which is 9
    cm long, 5 cm wide and 10 cm high. Calculate the area of the base of the
    larger cuboid.

3. A water tank is in the shape of a cylinder radius 2 m and height 3 m. A similar
    tank has a radius of 1.5 m. Calculate the height of the smaller tank.

4. Write down the dimensions of any two  cubes. Are the two cubes similar? Are all
    cubes similar? Are all cuboids similar?

5. A designer has two models of a particular car. The first model is 15
    cm long, 7.5 cm wide and 5 cm high. The second model is 3.75 cm long,
    1.70 cm wide and 1.25 cm high. He says that the two are ‘accurate scale
     models’. Explain whether or not his claim could be true.

11.2 Enlargement

11.2.1 Definition of enlargement
       Activity 11.9

1. Compare the shapes and looks of the two pictures. What do you
     notice?

2. By measuring, determine how many times picture (b) is bigger
    than picture (a)

3. What is the name of the transformation that transforms
    Fig. 11.43 (a) to (b)?

4. Identify the requirements for the transformation to be performed?
    Consider the Triangles in Fig.11.44.

The two triangles have exactly the same shape. However, triangle A′B′C′ is bigger
than triangle ABC. We say that the triangle ABC has been enlarged to triangle A′B′C′ .
Enlargement is the transformation that changes the size of an object but preserves
its shape i.e angles are preserved. Lines AA′, BB′ and CC′ produced meet
at a common point 0. The point is called the centre of enlargement. Triangle ABC
and A′B′C′ are similar. By measuring, determine the value of
OA′/OA
. This is called the scale factor of enlargement.

The scale factor of enlargement is defined as the ratio.

Note:
This particular scale factor is often called the linear scale factor because
it is the ratio of the lengths of two line segments.
Thus, the scale factor of enlargement (k) tells
us:
(a) How big or small the image is compared to the object.

(b) How far and in which direction is the image in respect to the object.
    Therefore, for an enlargement to be performed, the centre and scale factor of
     enlargement must be known.

11.2.2 Constructing objects and images under enlargement

11.2.2.1 Positive scale factor > 1

Activity 11.10

1. Draw any triangle ABC.

2. Taking a point O outside the triangle as the centre of
    enlargement, and with a scale factor 3, construct the image
    triangle A′B′C′.

The following is the procedure of constructing the image of triangle XYZ
under enlargement scale factor.

(a) Draw triangle XYZ and choose a point O outside the triangle.

(b) Draw construction lines OX, OY and OZ, and produce them (Fig. 11.45).

(c) Measure OX, OY and OZ. Calculate the corresponding lengths OX′,
     OY′ and OZ′; and mark off the image points. From the definition
     of scale factor, it follows that image distance = k × object distance,
     where k is the scale factor. Thus, OX′ = 3OX, OY′ = 3OY and
     OZ′ = 3OZ.

(d) Join points X′, Y′, Z′ to obtain the image triangle.

Example 11.15

Enlarge the triangle ABC given in the grid (Graph 11.1) by a scale factor 2
with centre of enlargement at (0,0).

11.2.2.2 Fractional Scale factor

      Activity 11.10

Construct the enlargement of triangle PQR shown in Fig. 11.51 with O as
the centre of enlargement and scalefactor 1/3.

We notice that:
Under enlargement with a scale factor
that is a proper fraction:

1. The object and the image are on the same side of the centre
    of enlargement.

2. The image is smaller than the object and lies between
     the object and the centre of enlargement.

3. Any line on the image is parallel to the corresponding line on the
    object.

Example 11. 17

   Enlarge triangle ABC with a scale factor1/2
,  centered about the origin.

OB′ =1/2OB
OC′ =1/2OC
Since the centre is the origin, we can in
this case multiply each coordinate by 1/2
to get the answers.

A = (2, 2), so A′ will be (1, 1).

B = (2, 6), so B′ will be (1, 3).

C = (4, 2), so C′ will be (2, 1).

11.2.2.3 Negative scale factor
       Activity 11.12

In this activity, you will locate that ΔA′B′C′ as the image of ΔABC in
Fig. 11.55 under enlargement centre O, L.S.F –3, locate the image.

Procedure

(i) Join each vertex of ΔABC (object points) to centre O with straight lines.

(ii) Prolong each of these lines in the opposite side from centre O.

(iii) Measure the distance of each object point to centre O.

(iv) Multiply each of the object distance by 3 to get the
     corresponding image distance.

(v) Using the image distances obtained in (iv) above, locate the
     image point of each object point on the opposite side from O on
     the prolonged lines.

(vi) Join the image points to form the required image triangle.
      Fig. 11.56 shows the required image. (ΔA′B′C′).

Consider Fig. 11.57, in which the centre of enlargement is O and both images of flag
ABCD are 1.5 times as large as the object.

A′B′C′D′ is the image of ABCD under enlargement with centre O and scale
factor 1.5. A′′B′′C′′D′′ is also an image of ABCD
under enlargement with centre O. How is it different from A′B′C′D′?
OA = 1.2 cm and OA′′ = 1.8 cm. OA and OA′′ are on opposite sides of O. If
we mark A′′OA as a number line with O as zero and A as 1.2, then A′′ is –1.8

(Fig. 11.58).

We say that the scale factor is –1.5 because
–1.5 × 1.2 = –1.8. We write OA′′ = –1.5 OA.
We notice that: Under enlargement with a negative
scale factor,

1. the object and the image are on opposite sides of the centre of
    enlargement,

2. the image is larger or smaller than the object depending on whether
    the scale factor is greater than 1 and negative or a negative proper fraction,

3. any line on the image is parallel to the corresponding line on the
    object, but the image is inverted relative to the object.

Example 11.18

Enlarge the rectangle WXYZ (Fig 11.5) using a scale factor of - 2, centered about
the origin.

11.2.3 Locating the centre of enlargement and finding the
           scale factor

           Activity 11.13

1. Fig. 11.61 show triangle ABC and its image triangle A′B′C′
     under the enlargement.

(a) Locate by construction the centre of enlargement

Example 11.19

Fig.11.62 shows a quadrilateral and its image under a certain enlargement.

(a) Locate the centre of enlargement.

(b) Find the scale factor of enlargement.

(c) Given that a line measures 5 cm,  find the length of its image under
    the same enlargement.

Solution

(a) To locate the centre of enlargement O, join A to A′, B to B′, C to C′
      and D to D′ and produce the lines (Fig. 11.61). The point where
      the lines meet is the centre of enlargement.

Note:
1. Given an object and its image, it is sufficient to construct only two
    lines in order to obtain the centre of enlargement.

2. An enlargement is fully described if the centre and scale factor of
    enlargement are given.

Exercise 11.8

1. In Fig. 11.64, ΔOA′B′ is the enlargement of ΔOAB.

    (a) Which point is the centre of enlargement?

    (b) Find the scale factor ofenlargement.

    (c) Calculate the lengths A′B′ and AA′.

2. In Fig. 11.65, rectangle PQRS is an enlargement of rectangle ABCD with
    centre O.

(a) Find the scale factor of enlargement.

(b) Which point is the image of point

(i) A;       (ii) B;         (iii) C?

(c) Find the length of the diagonal of rectangle PQRS given that the length
     of the diagonal of rectangle ABCD is 15 cm.
3. Fig. 11.66 shows a triangle XYZ and image triangle X′Y′Z′. Copy the two
     triangles and by construction find:

(a) centre of enlargement

(b) scale factor of enlargement

4. Fig. 11.67 shows triangle ABC and its image triangle ABC under an
    enlargement.

Copy the two triangles and by construction find:

(a) centre of enlargement

(b) scale factor of enlargement.

5. In a scale model of a building, a door which is actually 2 m high is
    represented as having a height of2 cm.

(a) What is the scale of this model?

(b) Calculate the actual length of a wall which is represented as
      being 5.2 cm long.

6. When fully inflated, two balls have radii of 10.5 cm and 14 cm respectively. They
    are deflated such that their diameters decrease in the same ratio. Calculate
    the diameter of the smaller ball when the radius of the larger ball is 10 cm.

7. A tree is 6 m high. In photographing it, a camera forms an inverted image
    1.5 cm high on the film. The film is then processed and printed to form a
    picture in which the tree is 6 cm high.
   Calculate the scale factors for the two separate stages.

8. Δ A′B′C′ is the image of Δ ABC (Fig. 11.68) after an enlargement.

(a) Find by construction the centre of enlargement.

(b) Find the scale factor of the enlargement.

(c) Given that a line segment measures 8.5 cm, find the
     length of its image under this enlargement.

11.2.4 Properties of enlargement
     Activity 11.14
1. Use reference materials, internet and other resources to research
   on the properties of enlargement.

2. State and explain the properties.

3. Where do you think the properties apply in real life?
  

  Having learnt how to enalarge figures
  using any scale factor, and to determine

  the centre of enlargement, confirm the following properties of enlargement as a
   transforming the activities we did.

1. An object point, its image and the centre of enlargement are collinear.

2. For any point A on an object, OA = kOA , where k is the scale
    factor.
(a) If k = 1, the object id mapped onto itself.

(b) If k > 0, the object and its image lie on the same side of
      the centre of enlargement.

(c) If k < 0, the object and its image are on opposite sides of
      the centre of enlargement.

3. The centre of enlargement is the only point that remains fixed
     irrespective of the scale factor.

4. Both object and image are similar

5. If the linear scale factor of enlargement is k, the area scale
    factor is k2. Where the enlargement is of a solid, the volume scale factor
     is k³.

11.2.5 Enlargement in the Cartesian plane
       Activity 11.15
1. On a squared paper, draw a quadrilateral with vertices A(0, 3),
    B(2, 3), C(3, 1) and D(3, –2).

2. Copy and complete table 11.1 for the coordinates of A′B′C′D′
    and P′, the images of points A, B, C, D and a general point P(a,

b) under enlargement with centre the origin and the given scale
    factors.

Note:
An enlargement with centre (0, 0) and scale factor k maps a point P(a, b) onto
P′(ka , kb).

Note:
It is not possible to generalise for P (a , b) when the centre is not the origin. When
the centre of enlargement is not the origin, we must carry out complete construction

Example 11.20

In a cartesian plane plot traingle ABC whose coordinates are A(1, 0), B(3, 2) and
(2, 4). On the same cartesian plane locate and draw triangle A′B′C′ the
image of triangle ABC under enlargement scale factor 3 and centre origin.

Solution

Using the steps we learnt earlier and also fact that an enlargment centre origin
scale factor K maps point P(a, b) tp P′(ka, kb) we obtain DA′B′C′ as shown in Fig. 11.7.

Example 11.21

The coordinates of a quadrilateral are Q(0, 1), R(2, 2) S(2, 4) and T(0, 3).
 (a) plot the quadrilateral QRST on cartesian plane. Identify the
      quadrilateral by name?

(b) By construction, locate and draw the image of quadrilateral QRST
      on the cartesian plane under enlargement scale factor -2 centre
     (-2). Name the image as Q′R′S′T′.

(c) Write down the co-ordinates of vertices of the image.

Exercise 11.9

1. Points A(–2, –1), B(1,–1), C(1, –4) and D(–2, –4) are vertices of a
   square. Without drawing, state the coordinates of the vertices of the
   image square under enlargement with centre origin and scale factor:
(a) –4      (b) –2     (c) –1    (d) 1/– 4

2. Triangle ABC with vertices A (9, 4), B
    (9, 1) and C (15, 1) is mapped onto
     Δ A′B′C′ vertices A′ (1, 2), B′ (1, 1)
     and C′ (3, 1) by an enlargement.
     Find:
    (a) the scale factor of the enlargement.

    (b) the co–ordinates of the centre of enlargement.

3. Triangle ABC has vertices A(1, 2),B (3, 2), C (3, 4). Find the co–
ordinates of the image Δ A′B′C′ after
an enlargement, centre (0, 0) scale
factor –3.

4. The points A (1, 1), B (3, 2),
C (3, 4) and D (1,3) are vertices of a quadrilateral.

(a) Draw the quadilateral on the Cartesian plane.

(b) Taking O(4, 0) as the centre of enlargement, find the image
      when the linear scale factor is:
(i) 1.5 (ii) –2.4

5. The vertices of an object and its image after an enlargement are
A (–1, 2), B (1, 4), C (2, 2) and A′ (–1, –2), B′(5, 4), C′(8, –2)
respectively.
Draw these shapes on squared paper.
Hence, find the centre and scale factor of enlargement.

6. On squared paper, copy points A, B, C, D and F as they are in graph 11.9.
    Given that ΔDEF is an enlargement of ΔABC, find the coordinates of E.
     What is the centre of enlargement?

7. Points A(4, 0), B (0, 3) and C (4, 3)are the vertices of a triangle. Draw
   the triangle on a squared paper. Copy and complete Table 11.3 for
   the co–ordinates of A′, B′ and C′; the images of A, B and C under
   enlargement with centre (0, 0).

8. The vertices of Δ ABC are A (3, 2), B (1, 4) and C (4, 4). Find the image
    of Δ ABC under enlargement, centre (0,0) and scale factor:
    (i) –2        (ii) 1/2
.
9. The vertices of figure ABCD are A(1, 1), B(2, 4), C(1, 5) and D(5, 4).
     Draw the figure and its image A′ B′ C′ D′ under enlargement scale factor +2
     centre (–2, 3).

10. The vertices of ΔABC are A(5, 2), B (7, 2) and C (6,1). Draw the
       triangle and its image ΔA′B′C′ under enlargement scale factor –3 centre
      (4, 1).

11. The vertices of a triangle are X(1, 2), Y(3, 1) and Z(3, 3) and the vertices
      of its image are X′(–4, 4), Y′(–8, 6) and Z′(–8, 2).

(a) Find the centre of enlargement and the scale factor.

(b) Using the centre of enlargement in (a) above, locate the image
     X′′Y′′Z′′ of XYZ using the scale factor of 2.5.

11.2.6 Area scale factor
       Activity 11.17

1. Draw a rectangle. Choose a point O anywhere outside the
     rectangle.

2. With O as the centre, enlarge the rectangle with scale factor 2.

3. What are the dimensions of the image rectangle?

4. Calculate the alinear scale factor (L.S.F) of the enlargement.

5. Calculate the areas of the two rectangles and the ratio

This ratio is called the area scale factor.

6. How does this ratio compare with the linear scale factor?

7. Repeat your enlargement but with scale factors 3, –2 and 1/2.

How do the new area scale factors compare with the corresponding
linear scale factors?

Area scale factor (A.S.F) = (linearscale factor)2.

∴ L.S.F = A.S.F

Example 11.22

The ratio of the corresponding sides of two similar triangles is
3/2. If the area of the smaller triangle is 6 cm2, find the
area of the larger triangle.

Exercise 11.10
1. A rectangle whose area is 18 cm2 is enlarged with linear scale factor 3.
   What is the area of the image rectangle?

2. A pair of corresponding sides of two similar triangles are 5 cm and 8 cm
    long.
(a) What is the area scale factor?

(b) If the larger triangle has an area of 256 cm2, what is the area of
     the smaller triangle?

3. The ratio of the areas of two circles is 16 : 25.

(a) What is the ratio of their radii?

(b) If the smaller circle has a diameter of 28 cm, find the diameter of the
      larger circle.

4. Two photographs are printed from the same negative. The area of one
     is 36 times that of the other. If the smaller photograph measures 2.5 cm
      by 2 cm, what are the dimensions of the larger one?

5. A lady found that a carpet with an area of 13.5 m2 fitted exactly on the
    floor of a room 4.5 m long. If she moved the carpet to a similar room
    which is 1.5 m longer, how much floor area remained uncovered?

6. An architect made a model of a building to a scale of 1 cm : 2 m. A
    floor of the building is represented on the model with an area of 12 cm2.
    What would be the corresponding area on another model of the same
    building whose scale is 2 cm : 1 m?

7. The scale of a map is 1:500 000. A section of sea has an area of 38.6
    cm2. Find the actual area of the sea represented on the map in hectares.

8. Small packets of tea leaves were packed in boxes measuring 30 cm by 42 cm
    by 24 cm for dispatch. Given that each packet was similar to the container
    and the ratio of sides of the box to the packet was 4 : 1, find the surface area
    of each packet of tea leaves.

11.2.7 Volume scale factor
     Activity 11.18

Fig. 11.69 shows a cuboid and its enlargement with scale factor 2.

This ratio is called the volume scale factor (V.S.F).

3. How does this ratio compare with the linear scale factor?

4. If instead the scale factor of enlargement is 3, what are the
    dimesions and volume of the image? How does the volume
    scale factor compare with the

linear scale factor now? If the scale factor of enlargement is1/2,
what is the volume scale factor?

5. Given area scale factor, how would one get volume scale
    factor; and vice versa,

Note: For solids of the same material, the ratio of their masses is equal to the
          ratio of their volumes.

Example 11.24

A solid metal cone with a base radius 7 cm has a volume of 176 cm3. What
volume would a similar cone made of the same metal and with base radius
10.5 cm have?

Example 11.25

Two similar cones A and B are such that the ratio of their volumes is 108:500. The
 smaller cone has a curved surface area of 504 cm2. Find the area of the curved
 surface of the bigger cone.

Example 11.26

The areas of two similar solids is 49 cm2 and 64 cm2.

(a) Find their volume scale factor.

(b) If the smaller one has a volume of 857.5 cm3,what is the volume of
      the larger one?

Exercise 11.11

1. The volume scale factor of two similarsolids is 27. What are their linear and
      area scale factors?

2. A cylinder with base radius r has a volume of 77 cm³. What is the volume
    of a similar cylinder with base radius
     (a) 2r (b) 3r ?

3. Two similar jugs have capacities of 2 litres and 3 litres. If the height of the
     larger jug is 30 cm, find the height of the smaller one.

4. Two similar cones made of the same wood have masses 4 kg and 0.5 kg respectively.
     If the base area of the smaller cone is 38.5 cm², find the base area of the larger one?

5. A concrete model of a commemoration monument is 50 cm high and has a mass
    30 kg.
(a) What mass will the full size monument of height 9 m have if it is also made of concrete?

(b) If the surface area of the model is 5000 cm², what is the surface area of
  the full size monument?

6. Fig. 11.70 is a drawing of a model of a swimming pool. Find the capacity of the
    actual pool, in litres, given that its length is 27 m.

7. The volume of two similar cones are 960 cm³ and 405 cm³. If the area of the curved
     surface of the bigger cone is 300 cm², find the surface area of the smaller cone.

8. The surface areas of two similar containers are 810 cm2 and 1 440 cm2. Find the
    linear scale factor of the containers. Hence, find the capacity of the larger
    container if the smaller one has a capacity of 108 litres.

9. Two similar square based pyramids have base areas of 9 cm² and 36 cm²
     respectively. Calculate
     (a) the height of the larger pyramid if the smaller pyramid is 9 cm high.

     (b) the ratio of the height to the width of the smaller pyramid if the ratio
          for the larger pyramid is 3 : 1.

(c) the inclination of a slant face of the larger pyramid if for the smaller one
     is 76°.

(d) the volume of the larger pyramid if the smaller one has a volume of 27 cm3.

Unit Summary

• Two triangles are similar if the corresponding sides are in proportion,
    i.e. have a constant ratio or the corresponding angles are equal.
    Congruent triangles are also similar.

• For all shapes, other than triangles, both similarity conditions must be satisfied
   for them to be similar.

•. The ratio of the corresponding sides in similar figures is called a linear scale
     factor.

• For all solids, corresponding angles must be equal and the ratios of corresponding
   lengths must be equal for the solids to be similar.

• All cuboids are equiangular since all the faces are either rectangular or squares.
   However, not all cuboids are similar but all cubes are.

• If two figures are similar, the ratio of their  corresponding areas equals the square of
   their linear scale factor. A.S.F = (L.S.F)²

• If two solids are similar, then the ratio of their corresponding volumes equals the
   cube of their linear scale factor. V.S.F = (L.S.F)³

• An enlargement is defined completely if the scale factor and the centre of
 enlargement are known.

• In an enlargement the object and its image are similar. If the scale factor is
   ±1, the object and its image are identical. Scale factor of enlargement

• To locate the centre of enlargement given an object and its image,

(a) join any two pairs of corresponding points.

(b) produce the lines until they meet at a point, that point is the centre of
      enlargement.

• An enlargement scale factor k centre origin (0, 0) maps a point P(a, b) onto
   P′(ka , kb).

Unit 11 test

1. In ΔABC (Fig. 11.71) identify two similar triangles in the figure and use them to
    find the values of a and b

2. ΔABC has coordinates A(–3, 1), B(–3, 4), C(–1, 4). ΔA′B′C′ is such that A′(–1, –3),
    B′(–1, 3) and C′(3, 3). Draw the two triangles on a graph paper. Find the
    centre and the scale factor of enlargement that maps ΔABC onto ΔA′B′C′.

3. Find the image of ΔABC A(2, 4), B(4, 2), C(5, 5), under enlargement centre (0, 0)
    scale factor 2 and state the coordinates

     of A′, B′ and C′, the images of points A, B and C.

4. The volumes of two similar cuboids are 500 cm3 and 108 cm³ respectively. Find
    the ratio of their

(a) volumes, (b) surface areas. If the larger cuboid has a total surface
     area 400 cm2, find the surface area of the smaller cuboid.

5. In ΔABC, BC = 5 cm, AC = 6 cm and ∠ACB = 49°. If ΔABC is enlarged to
    ΔA′B′C′ using a linear scale factor 3/2 ,

    (a) write down the size of ∠A′C′B′ in relation to ΔABC

    (b) calculate the length of the side A′C′.

    (c) calculate the ratio of the area of ΔABC to the area of ΔA′B′C′ in
         relation to DABC.

(d) find the ratio A′C′ : B′C′.

6. In the Fig. 11.72, ABCD is a trapezium with AB parallel to DC. The diagonals
    AC and BD intersect at E.

(a) Giving reasons, show that ΔABE is similar to ΔCDE.

(b) Given that AB = 3DC find the ratio DB to EB.

7. A bowl in the shape of a hemisphere has a radius of length 10 cm. A similar bowl
    has a radius of length 20 cm. Calculate:

(a) the circumference of the larger bowl if that of the smaller one is 64 cm.

(b) the surface area of the larger bowl if that of smaller one is 629 cm2.

(c) the capacity of the larger bowl if that of the smaller one is 2.1 litres.

8. In Fig. 11.73, AB DE. Identify two similar triangles and use them to find the
     values of x and y.