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Chapter 7 Introduction
May 11, 2013
Chapter 7 Introduction
Refer to pages 326-329 in the textbook.
Equivalent Ratios
A ratio is a relationship between two or more numbers expressed in terms of division (note: if
the numbers represent measurements, the measurements must all be in the same units). For
example, if there are 6 apples and 3 oranges in a bowl of fruit, we would say that the ratio of
apples to oranges is “6 to 3”. This ratio can be written as 6:3; however ratios, like fractions, are
normally written in lowest terms, by dividing out any factor common to all the numbers, so in
ଶ
this example, we would write the ratio as 2:1. A ratio may also be written as a fraction, e.g.
ଵ
(which is always expressed as a proper or improper faction, never as a whole or mixed number).
If two ratios are equal, we can write an equation expressing this as (for example) 6:3 = 2:1, or
଺
ଶ
(more commonly) in fraction notation as ଷ ൌ ଵ . Such an equation is called a proportion.
଻
ଷ
Frequently proportions contain an unknown; for example, ൌ . To solve a proportion means
௭
ଶ
(as it does for equations) to find the value of the unknown which make the proportion true.
Proportions are most easily solved by cross-multiplication. Given a proportion of the form
ܽ ܿ
ൌ
ܾ ݀
we can multiply the numerator of each side with the denominator of the other side, equating the
results to get
ܽ݀ ൌ ܾܿ
଻
ଷ
Applying this to the example ௭ ൌ ଶ , we get 3z = 7×2, so 3z = 14 and z =
ଵସ
ଷ
.
Transformations
The text introduces two new transformations, rotation and dilation.
A rotation moves every point on an object through an arc of a given size (e.g. 45°), called the
angle of rotation, each arc having the same point, the centre of rotation, as its centre. The
resulting image has the same size and shape as the original object, i.e. it is congruent to the
original object. Using a different centre of rotation changes only the position of the image, not
its orientation.
A dilation multiplies the distance between each point on an object and a common point, called
the dilation centre, by the same factor, the dilation factor. A dilation factor greater than 1
enlarges the resulting image; a dilation factor between 0 and 1 shrinks the image. The image has
the same shape as the original, i.e. it is similar to the original object. Using a different dilation
centre changes only the position of the image, not its size.
MPM2D—S. Inrig/A. Hamilton
Page 1 of 2
Chapter 7 Introduction
May 11, 2013
Similar and Congruent Triangles
Definition: two figures are similar if they have the same shape. In similar polygons, each
interior angle in one is equal to the corresponding angle in the other, and the length of each side
of one is the same multiple of the length of the corresponding side of the other. The constant
multiple is called the scale factor.
Conditions for similar triangles: two triangles are similar if
any of the following is true:
A
1
2
• SSS – all three pairs of corresponding sides are
proportional (i.e., related by the same scale factor)
• SAS – two pairs of corresponding sides are
proportional, and the contained angles (the angles between
these sides in each triangle) are equal
• AA – two of the angles of one triangle are equal to
two angles of the other.
3
C
B
X
1
Y
2
3
Z
The symbol “~” means “is similar to”. So we can state that
the two triangles on the left are similar by writing
∆XYZ ~ ∆ABC.
Note that when stating that triangles are similar, the order of the vertices is important: the first
named vertex of one must correspond to the first named vertex of the other, and so on. So ∠X
corresponds to (and is equal to) ∠A, ∠Y corresponds to ∠B, etc.
Definition: two figures are congruent if they have the same shape and size. In congruent
polygons, corresponding interior angles are equal, and corresponding sides are equal.
Conditions for congruent triangles: two triangles are congruent if any of the following is true:
•
•
•
SSS – all three pairs of corresponding sides are equal
SAS – two pairs of corresponding sides are equal, and the contained angles (the angles
between these sides in each triangle) are equal
ASA – two of the angles of one triangle are equal to two angles of the other, and the sides
between the two angles in each triangle are equal.
The symbol “؆” means “is congruent to”. The rule stated above for naming similar triangles
also applies when naming congruent triangles. If we have two congruent triangles PQR and
STU, we can write ∆PQR ؆ ∆STU provided that ∠P corresponds to ∠S, and so on.
MPM2D—S. Inrig/A. Hamilton
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