cares is written bikes:cars = 10:20, which reduces to 1:2. This is said “one
to two,” meaning that for every one bike I have, you have two cars. This
ratio can also be written as ^1200 = ^12 .
A proportionis a statement that compares two equal ratios. Maybe the cur-
rent ratio of my blue pens to my black pens is 7:2 or ^72 . If I add four more
black pens to my collection, a proportion can be used to determine how
many blue pens I must add to maintain the same ratio of blue pens to black
pens in my collection:
bbllaucekppeennss= ^72 = 2+p 4 = 6 p
Since the new denominator of six black pens is three times bigger than
the original denominator of two black pens, I must multiply the seven blue
pens in the numerator by three also, in order to maintain the same ratio.
Therefore, I will need to have 21 blue pens, or 14 more blue pens than I had
originally, if I want to maintain the same ration of blue to black pens. Notice
that ^261 reduces to ^72 , which was the original ratio.
Ratios and Proportions in Similar Triangles
If two triangles are similar, then the triangles will be proportional. What this
means is that the ratios and proportions of their corresponding sides will be
equal. This is useful because when dealing with similar triangles, propor-
tions can be used to solve for unknown sides in the triangles. In the illus-
tration below, ΔABC is similar to ΔEDF and the symbol that is used to
represent that is ≈. Therefore ΔABC ≈ΔEDF. Based on the congruencies
of the pairs of angles B and E, and C and F, it is clear that BCis proportional
to EFand also that BAis proportional to ED. With this information a pro-
portion can be written as follows:
BBCA
=
E
ED
F
^160 = 1 x 2 Since 12 is twice as large as six, it follows that xwill have to be twice as large
as 10, so x= 20. If DFwas 18 units long, it would follow that ACwould be
nine since ΔABC is half as big as ΔEDF.
501 Geometry Questions