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Similarity and enlargement
Map scale is the relationship between a unit of length on a map and the corresponding
length over the ground*.
A typical verbal scale might be “One inch to one mile”. Many maps carry a graphic
scale such as this bar scale.
A representative fraction (RF) shows the relationship between one of any unit on the
map and the same units on the ground. RFs may be shown as an actual fraction (e.g.
1/25,000). They are more usually written like a mathematical proportion with a colon
(as in 1:25,000).
Example:
1:25,000 means that one unit of any length on the map
represents 25,000 of the same units on the ground
The RF is the “scale factor” of the supposed enlargement between the map and the
ground.
LENGTH ON THE MAP = LENGTH OVER THE GROUND · RF
The RF is versatile because it is not tied to any specific unit of measure. You may work
in any unit you choose - metric, US or any other.
What do we mean by Small Scale and Large Scale?
Cartographers use these terms differently to normal English usage. Cartographers are
referring to the relative size of the representative fraction. A large scale map is where
the RF is relatively large. An RF of 1:25,000 (1/25,000) is larger than an RF of
1:1,000,000 (1/1,000,000). A large scale map uses more map area to describe a given
area on the ground than a small scale map. Many people are initially confused by this
terminology as the small scale map shown a much greater area of the ground than a
large scale map of the same size.
Map scales help us measure distance and area of and between the real world objects
represented on the map.
* True distance along the ground may be influenced by the topology of the ground in
question. Map scale is the relative difference between a unit length on the map and the
corresponding distance as the crow flies (even if the crow has to fly very high and
level).
An angle of particular significance is the right-angle: the angle at each corner of a
square or a rectangle. Indeed, a rectangle can always be divided into two triangles by
drawing a line through opposing corners of the rectangle. This pair of triangles has
interesting properties. First, the triangles are identical. Second, they each have one angle
which is a right-angle.
Similarity of triangles
Angle-Angle (AA) theorem
If two triangles have two pairs of corresponding angles equal, then the triangles are
similar.
Aˆ  Aˆ ' and Bˆ  Bˆ '  ABC and A’B’C’ are similar
Side-Side-Side (SSS) theorem
If two triangles have all three pairs of corresponding sides in the same ratio,
a ' b' c '
   ABC and A’B’C’ are similar
then the triangles are similar.
a b c
Side-Angle-Side (SAS) theorem
If two triangles have one pair of corresponding angles equal and both corresponding
pairs
of sides adjacent to the angle have the same ratio, then the triangles are similar.
b' c'
Aˆ  Aˆ ' and   ABC and A’B’C’ are similar
b c
Further comsequences:
When two right-angled triangles have one pair of corresponding acute angles
equal
the triangles are similar.
The height perpendicular to the hypothenuse of a right-angled triangle splits it
into
two triangles that are similar to the original one.