Middle Years Programme (MYP) Mathematics: Similar Shapes Investigation

This document presents a comprehensive solution to a Grade 9 mathematics assignment focusing on similar shapes. The assignment explores the properties of two-dimensional (2D) and three-dimensional (3D) objects, defining key concepts such as dimensions, vertices, edges, and faces. It investigates the relationships between 2D and 3D objects and the characteristics of similar shapes, including the concept of scale factor. The solution provides detailed explanations and calculations for tasks involving mathematically similar shapes, rectangles, and triangles, determining their similarity based on ratios of corresponding sides and angles. Furthermore, the document analyzes the relationships between 3D objects, particularly cylinders and spheres, calculating and comparing their volumes and surface areas using the scale factor. The solution includes clear step-by-step workings and references relevant mathematical principles, providing a complete and insightful understanding of the topic.

SIMILAR SHAPES

Similarity 1
Two dimensional (2D) objects
Two dimensional objects are objects which only have two dimensions, for instance width and length
with the width missing. 2D objects can be completely described using two dimensions. A dimension can
be defined as a quantity whose magnitude can be measured, for example the height of an object. A two
dimensional object is flat and it is completely defined by the dimensions of its sides and the angles
between those sides. Sometimes, the angles on the angles are known as vertices.
Three dimensional (3D) objects
On the other hand, a three dimensional object has three dimensions. This is the main distinguishing
feature between two dimensional and three dimensional objects ("2D/3D Transformations," 2014). In
addition to the two dimensions of length and width present on a two dimensional object, a three
dimensional object has one additional dimension of thickness or depth. Even if the depth is very small,
the object still fits the description of a 3D object. All the real objects we observe around us are three
dimensional objects. For example, buildings, vehicles, utensils and many more.
The connection between 3D and 2D objects.
There exists several relationships between two dimensional and three dimensional objects. A three
dimensional object can be said to be made of several two dimensional objects. For example, we can
visualize two dimensional surfaces on the faces of three dimensional objects. For instance, a cube has
four flat (2D) surfaces.
Properties of 2D objects
For this part, we investigate the properties of a two dimensional object such as a square.
Consider the object above. This is a two dimensional object drawn on a two dimensional surface. From
the figure, we observe the following.
 It has four sides of equal dimensions.
 All the angles at the vertices are equal at 90 degrees each.
 The four sides are perpendicular to each other.
 The sides opposite to each other are parallel. This means that even if the sides are extended
indefinitely, they never meet anywhere.

Similarity 2
 All the angles at the vertices add up to 360 degrees. For instance, for the square above the sum
of the angles at the vertices is : 90 + 90 +90 +90 =360 degrees (or equivalently, the sum = 90 × 4
= 360 degrees)
 The lines drawn from the vertices through the body are called diagonals and they intersect each
other at 90 degrees. They are also equal in magnitude.
Properties of 3D objects
Three dimensional objects are more complex compared to two dimensional objects. In the case of 3D
objects, we talk about edges, vertices and faces. Consider the three dimensional figure shown below for
example.
The three dimensional figure is a cube which has the following properties:
 The faces of the figure are defined as the flat parts (an example is the shaded part). A cube has
six faces.
 Edges are defined as the lines where any two faces of a three dimensional object meet. This
figure has a total of twelve edges.
 Vertices are defined as the points where at least two edges meet. The figure above has a total of
eight vertices.
Similar shapes
Similar shapes are generally defined as figures with the same shape but different dimensions. In specific,
the corresponding angles in two similar shapes or figures are equal. The dimensions of the two similar
figures normally differ by a constant factor commonly known as the scale factor (Glaeser, 2017). The
scale factor is defined as the ratio of any two corresponding dimensions of the two similar figures. As
shown in task three, the scale factor is very important in the determination of the properties of three
dimensional objects. The scale factor can significantly simplify the calculations of quantities such as the
volumes ad surface area of similar three dimensional objects. There, it is shown that, knowing one
quantity such as the volume of one of the objects, we can easily determine the same quantity for the
other figure by simply using the scale factor.

Similarity 3
When carrying out enlargements, the magnitude of the scale factor indicates how big or how small an
object will be after enlargement (Rangelov, 2004). The scale factor can be a fraction which then means
that the resulting object after enlargement will be smaller than the original object. On the other hand, a
scale facto with a whole value indicates that the resulting object after enlargement will be larger than
the original objet.
Task 1: mathematically similar shapes.
a)
The relationships between the above right angled triangles are:
 All corresponding angles are equal ("Similar triangles," 2015)
 The ratio of corresponding sides is a constant
For example 6
3 = 8
4 =constant
 The corresponding angles are equal.
b)

Similarity 4
The two figures above are similar. One figure is simply the enlargement of the other. Alternatively, the
larger image has been scaled down to fit the smaller image. Since an enlargement only changes the
dimensions of the lengths, the corresponding angles will remain equal throughout the enlargement.
Task 2:
The two rectangles are similar. This is because the ratio of corresponding sides is a constant as the
relationship below shows.
12
9.6 = 8
6.4 =1.25=constant
b)
The ratios of corresponding sides are:
7.5
5 =1.5
3
1.5 =¿2
6
4 =1.5
Clearly, the two triangles are not similar since the ratios of corresponding sides are not constant as the
calculation above shows.

Similarity 5
Task 3:
Relationships between 3D objects.
For a cylinder:
Volume
Consider two cylinders of the same height h and different radii r1 and r2 respectively,
The volume of the first cylinder, V1 = π r12h
The volume of the second cylinder, V2 = π r22h
The ratio of the two volumes is thus, V 2
V 1 = π r2
2 h
π r1
2 h
Therefore, V 2
V 1 = r2
2
r1
2 and thus V2 = r2
2
r1
2 V 1
Therefore, the volume of the second cylinder is just the volume of the first cylinder multiplied by the
ratio of the squares of the radii of the two cylinders as shown above.
Surface area
The surface area of the first cylinder will be: S.A1 = 2πr12 + 2πh
The surface area of the second cylinder will be: S.A2 = 2πr22 + 2πh
The ratio of the two areas will be: S . A2
S . A1
= 2 π r2
2 +2 πh
2 π r1
2 +2 πh
For a sphere
Volume
Consider two spheres of radii r1 and r2 respectively:
The volume of the first sphere is: V1 = 4 π r1
3
3
The volume of the second sphere is: V2 = 4 π r2
3
3
The ratio of the two volumes is thus: V 2
V 1
= 4 π r 2
3
3 × 3
4 π r1
3 = r2
3
r1
3
Hence V2 = r2
2
r1
2 V1

Similarity 6
Therefore, the volume of the second sphere is just the volume of the first sphere multiplied by the
square of the ratio of the radius of the second sphere to that of the first sphere.
Surface area
The surface area of the first sphere is given by: S.A1 = 4πr12
The surface area of the second sphere is given by: S.A2 = 4πr22
The ratio of the two areas is then: S . A2
S . A1
= 4 π r2
2
4 π r1
2 = r2
2
r1
2 which is similar to the case for volume.

Similarity 7
References
2D/3D Transformations. (2014). Mathematical Techniques in GIS, Second Edition, 205-222.
doi:10.1201/b16910-11
Rangelov, D., 2004. Types of geometrical transformations and perceptual similarity of
figures. Psihologija, 37(4), pp.483-493.
Similar triangles. (2015). Plain Plane Geometry, 81-103. doi:10.1142/9789814740456_0004
Glaeser, G. (2017). Proportions and similar objects. Math Tools, 67-118. doi:10.1007/978-3-319-66960-
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