10. Similarity of Triangles - Equivalence of Two Criteria
In secondary mathematics, the four declarations
-
all circles are similar,
-
all squares are similar
-
rectangles with equal side aspect ratios are similar, and
-
triangles with corresponding sides proportional and corresponding
angles equal
may be given but not explained.
Experience with or assumptions about scaling in maps implies if in a
given correspondence between the vertices of two triangles, if the
coordinates of corresponding vertices are proportional then corresponding
sides are proportional and corresponding angles are equal.
The aim is to show the converse:
In two triangles, if corresponding sides are proportional and
corresponding angles are equal then there coordinate systems, one per
triangle, in which the coordinates of corresponding vertices are
proportional.
Proof of the Converse
Suppose between triangles  ABC and DEF,
there is a correspondence of vertices
With respect to this correspondence, suppose corresponding angles are
equal, and corresponding sides are proportional. Then the fractions or
ratios
have a common value K. Assume the perpendicular from A to the line
through through the other two vertices B and C falls between those
vertices as points of the line. Further assume the perpendicular from D
to the line through through the other two vertices E and F falls between
those vertices as points of the line. The foregoing assumptions are
depicted in the follow diagram.
Coordinates for ABC
Select or impose a coordinate system on ABC with the origin at the intersection of the perpendicular
with line BC, with the line segment B to C being along the horizontal
(first coordinate or abscissa) axis. That leads to coordinates for the
three vertices as shown.
Introduce the shorthand or function notation
.gif)
Now the Pythagorean theorem applied to triangles AOC and AOB gives
h2= b2-x2 =
c2-(x-a)2
Therefore (x-a)2 -x2 = c2 -
b2 and hence 2ax + a2 = c2 -
b2
The latter implies -2ax = c2 - b2 - a2 =
c2 - (a2 + b2)
Therefore the abscissa of point C is
-x = [c2 - (a2 + b2)]/2a = P(a,b,c) -
formula (1)
Now h2= b2-x2 implies the ordinate of
point A is
 - formula (2)
Finally, the abscissa of point C is
 -formula (3)
Formulas (1), (2) and (3) express the coordinates of vertices A, B and C
in terms of side lengths a, b and c.
Coordinates for DEF
For triangle DEF
like reasoning gives the following coordinates
Proportionality of Corresponding Sides implies Similarity
The proportionality assumption
implies (algebraic exercise for reader) that K times the coordinates of
D, E and F respectively yield the coordinates of A, B and C.
Therefore the vertices of the two triangle form similar sets of points.
Whence the corresponding triangles are similar in accordance with our
coordinate characterization or definition of similar sets of points.
End-Notes: The above formulas for the coordinates would not
change if the perpendicular from a vertex A to the opposite side
intersected the line through vertices B and C outside of the line
segment B to C in the following manner.
 (second case)
The above formulas for the coordinates would not change if the
perpendicular from a vertex A to the opposite side intersected the line
through vertices B and C outside of the line segment B to C in the
following manner. However the formula for x will change - be multiplied
by -1. .
 (third case)
Replacing x by -x in the above diagram would lead to formulas identical
to those implied above for ABC in the first two cases depicted.
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Work Booklets for ages 3+ to 13 Use these or others to check
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children. The selection acquired in Canada is published in the
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the choice is theirs. But in retrospect, the selection does not
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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