7.3. Similar Polygons and Scale Factors http://www.ck12.org
triangles are similarandall squares are similar.If two polygons are similar, we know the lengths of corresponding
sides are proportional. In similar polygons, the ratio of one side of a polygon to the corresponding side of the
other is called thescale factor. The ratio of all parts of a polygon (including the perimeters, diagonals, medians,
midsegments, altitudes) is the same as the ratio of the sides.
Example A
Suppose 4 ABC∼ 4JKL. Based on the similarity statement, which angles are congruent and which sides are
proportional?
Just like in a congruence statement, the congruent angles line up within the similarity statement. So,^6 A∼=^6 J,^6 B∼=
(^6) K,and (^6) C∼= (^6) L. Write the sides in a proportion: ABJK=BCKL=ACJL. Note that the proportion could be written in
different ways. For example,ABBC=KLJKis also true.
Example B
MNPQ∼RST U. What are the values ofx,yandz?
In the similarity statement,^6 M∼=^6 R, soz= 115 ◦. Forxandy, set up proportions.
18
30
=
x
2518
30
=
15
y
450 = 30 x 18 y= 450
x= 15 y= 25Example C
ABCD∼AMNP. Find the scale factor and the length ofBC.
Line up the corresponding sides,ABandAM=CD, so the scale factor is^3045 =^23 or^32. BecauseBCis in the bigger
rectangle, we will multiply 40 by^32 because^32 is greater than 1.BC=^32 ( 40 ) =60.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for more content.CK-12 Foundation: Chapter7SimilarPolygonsandScaleFactorsB
Concept Problem Revisited
All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball diamond. This means all the
sides are in a^9060 =^32 ratio. All the angles in a square are congruent, all the angles in both diamonds are congruent.
The two squares are similar and the scale factor is^32.