SIMILARITY, RATIOS, and PROPORTIONS

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SIMILARITY, RATIOS, and PROPORTIONS

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Suppose you have a triangle that you'd like to enlarge.
That is, you want to keep the shape exactly the same, but you want it to be bigger.
Perhaps you want an enlarged copy where the length of each side is three times the size of the original, as shown below:

Or, perhaps you have a quadrilateral that you'd like to reduce.
That is, you want to keep the shape exactly the same, but you want it to be smaller.
Perhaps you want a reduced copy where the length of each side is half the size of the original, as shown below:

In both cases, you are keeping the angles exactly the same,
and you are multiplying all the sides by an appropriate scaling factor.
When you made it three times as big, the scaling factor was 3 .
When you made it half as big, the scaling factor was 0.5 .

This idea of keeping the shape the same, but changing the size is made precise by the concept of similarity.
Very roughly, two geometric figures are said to be similar when they have the same shape, but not necessarily the same size.

To make the concept of similarity precise, we first need to review ratios and proportions.

DEFINITION: RATIO
Let a and b be real numbers, with b&neq;0 .
The ratio of a to b is the quotient ab .

A ratio automatically brings a scaling factor into the picture, as follows.
Give the ratio of a to b a simpler name, s , so that
ab := s .
(The notation := is frequently used in mathematics to mean equal, by definition.)

Then, a=sb .
So, b has been scaled by s to give a .
If s=3 , then a=3b , so that a is three times bigger than b .
If s=0.5 , then a=0.5b , so that a is half as big as b .

When we scale geometric figures, equal ratios automatically enter the picture.
Consider the picture below, where a quadrilateral has been scaled by a factor of s (here, s is greater than one) to get a new quadrilateral:

Notice that A=sa and B=sb and C=sc and D=sd .
Solving for s in each equation gives an equality of four ratios:
Aa = Bb = Cc = Dd := s .
Thus, equality of ratios arises very naturally in any scaling situation.
An equality of ratios is called a proportion:

DEFINITION: PROPORTION
A proportion is an equality of ratios.
Therefore, a proportion takes the form:

Aa = Bb

Using just a little bit of algebra, many useful equalities arise from proportions:

THEOREM: EQUIVALENT PROPORTIONS
Let a , b , A , and B be nonzero real numbers.
Then, the following are equivalent:

Aa = Bb

aA = bB

ab = AB

ba = BA

Ab=aB

Think about these equalities in the context of the scaled quadrilateral above!

Some of the equalities, like Aa = Bb and aA = bB , compare ratios between the two triangles.

Some of the equalities, like ab = AB and ba = BA , compare ratios within each triangle!

DEFINITION: CROSS-MULTIPLYING
The process of going from the proportion Aa = Bb to the equivalent statement Ab=aB is called cross-multiplying.

DEFINITION: the phrase corresponding sides are proportional
When a triangle with sides a , b , and c is scaled to get a new triangle with corresponding sides A , B , and C ,
then the phrase corresponding sides are proportional is used to describe the equality of ratios:

Aa = Bb = Cc
This same terminology is used when scaling any polygon.

Finally, we are ready for the precise definition of similar polygons:

DEFINITION: SIMILAR POLYGONS
Two polygons are similar if and only if
there exists a correspondence between their vertices such that
corresponding sides are proportional and corresponding angles are equal.

In general, to prove that two polygons are similar, you must check two different things:

that angles are equal; and
that corresponding sides are proportional.

It is an extremely useful result that for triangles (and triangles alone!) it suffices to check only that angles are equal!
Of course, if two angles are equal, then the third angles must be equal, and so we have the (unproved) theorem:

THEOREM: AA SIMILARITY THEOREM
Two triangles are similar if and only if
two angles of one triangle are equal to two angles of the other triangle.

This is NOT true for polygons other than triangles!
Look at the pictures below.
All the angles are the same, but the polygons are not similar!

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