Geometry
Date: ______
§12.6 Fundamental Theorem of Similarity
Theorem: If G ~ G‛ and k is the ratio of similitude, then…
(1) Perimeter (G‛) = k perimeter (G)
(2) Area (G‛) = k2 area (G)
(3) Volume (G‛) = k3 volume (G)
Fundamental Theorem of Similarity: If two figures are similar with
ratio of similitude k, then…
(a) corresponding <'s are equal
(b) corresponding lengths and perimeters are in the ratio k
(c) corresponding areas and surface areas are in the ratio k2
(d) corresponding volumes are in the ratio k3.
Examples:
1.
2.
RSTU ~ WXYZ. Give the ratios of the perimeters and the areas of these rectangles.
4
Hexagons ABCDEF and UVWXYZ are similar with ratio of similitude . If ABCDEF
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has area 56 square units, what is the area of UVWXYZ?
3.
4.
Two Toy trucks are shown. Each is similar to the same type of truck, so they are
similar to each other. The larger toy is 18 cm long, and the smaller is 6 cm long.
a)
How do the heights of the toy trucks compare?
b)
How do the areas of the floors of the holds (where the cargo is stored) compare?
c)
How do the volumes of the holds compare?
Before a wooden shed is built, a model of the shed is to be made. The size of the
model is to be 1 of the actual size.
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a)
How does the total length of wood needed for the model compare to the total
length of wood needed for the actual shed?
b)
How does the inside volume of the model compare to the inside volume of the
actual shed?
c)
If the wood needed for the model weighs 5 pounds, what is the weight of the
wood needed for the actual shed?
5.
The lengths of the sides of ∆ABC are 12, 15, and 17. The length of the shortest
side of a similar triangle, ∆DEF, is 5.
a)
What is the perimeter of ∆DEF, the similar triangle?
b)
How do the areas of the triangles compare?