Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
Chapter 6: Similar Triangles
Topic 1: Ratios of Similitude
RECALL:
In order for shapes to be congruent, there must be a ______________ _________________ that maps one
figure to the next.
Dilations are not rigid motions because the size of the figure changes. However, in a dilation, the measure of
all _________________ is still the same in both figures.
Similarity:
Two polygons are similar if and only if:
•
____________________________________________________________________
•
____________________________________________________________________
Once we know that two polygons are similar, we also know some additional facts about the figures.
1. All three corresponding sides are in proportion.
2. All three corresponding angles are equal.
The ratio of the lengths of the corresponding segments (or sides) is called the
______________________________________________________________
We must always show this ratio when answering questions of this type.
Name: _____________________________________________________________
Date: _________________________
Examples:
1)
Period: ______
6-1
What is the ratio of similitude?
Using this ratio, what is the length of
2) Given
is similar to
what is the longest side of
Using this ratio, what is the length of
, and the sides of
are
and
If the shortest side of
is 15,
.
3) If Tom is 6 foot tall and casts a shadow of 3 feet, while at the same time Jesse casts a shadow of only 2
feet, how tall is Jesse?
4) The ratio of similtude of
of
.
to
is
If the shortest side of
is 5, what is the shortest side
Name: _____________________________________________________________
Date: _________________________
Similarity in Dimensions
Period: ______
6-1
Not only are the _______________ of similar figures in ___________________, there are special relationships between
the _______________, _______________, and _______________ of similar figures as well.
1. Ratio of the Perimeters of Similar Figures:
Perimeter: Distance around the outside, expressed as units
The ratio of the ____________________ of similar figures is ____________ to the ratio of similarity (the ratio of
the sides). The ratio of the perimeters is also known as the _____________________________________.
2. Ratio of the Areas of Similar Figures:
Area: The total space inside of a flat figure, expressed as units squared
To find the ratio of the _____________ of similar figures we must _______________ the ratio of similitude.
3. Ratio of the Volume of Similar Figures:
Volume: The amount of space inside of a 3-D figure, expressed as units cubed
To find the ratio of the ____________________ of similar figures we must _________ the ratio of similitude.
Make a Chart!!!
A chart helps you keep all of your work neat and organized. It is very easy to quickly pick out the ratio you
need from the chart.
Examples:
1) Find the similarity ratio, the ratio of the perimeters, and the ratio of the areas of the two triangles below.
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
2) Find the ratio of the volumes of two similar polygons if the ratio of the perimeters is .
3) Find the ratio of the volumes of two similar cubes if the ratio of areas is
.
4) Find the ratio of the lengths of pairs of corresponding sides if the ratio of the areas of two similar
rectangles is 49:81.
5) If two similar boxes have widths of 8 inches and 5 inches, what is the ratio of their surface areas? What is
the ratio of their volumes?
6) Find the ratio of the volumes of two similar pyramids if the ratio of their surface areas is 121:169.
7) Find the ratio of the surface areas of two similar cones if the ratio of their volumes is 64:512.
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
Sometimes, instead of being asked to find the ratio of the perimeters, areas, or volumes, we are asked
to find the perimeter/area/volume for a specific shape. To do this, we must find the given similarity
ratio and then set up a __________________________.
8. The ratio of the surface areas of two similar prisms is 25:36. What is the perimeter of the smaller
prism if the perimeter of the larger prism is 56 cm. Round to the nearest tenth, if necessary.
9. The ratio of the volumes of two similar spheres is 125:512. Find the surface area of the larger
sphere if the surface area of the smaller sphere is 3200 m2. Round to the nearest tenth, if necessary.
10. The length of the sides of two triangles are in the ratio of 6:7. If the area of the larger triangle is
45 cm2, find the area of the smallest triangle to the nearest hundredth.
Name: _____________________________________________________________
Date: _________________________
Topic 1 Homework: Ratios of Similitude
Period: ______
6-1
Directions: Complete the following questions completely. Make sure to show all work and ratios of similitude!
If there is an area or volume formula that you have forgotten, ask Google!
1. Given
is similar to
is the longest side of
and the sides of
are 4, 7, 11. If the shortest side of
is 10, what
2. Amanda is 5 foot tall and casts a shadow of 4 feet. If Erin is 4.5 feet tall, how long is her shadow?
3. The sides of a quadrilateral measure 12, 18, 20 and 16. If the shortest side of a similar quadrilateral
measures 3, find the measures of the remaining sides of the quadrilateral.
4. If the sides of a triangle measure 4, 9, and 11, and the shortest side of a similar triangle measures 20, find
the measures of the remaining sides of this triangle.
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
5. In the figure,
a) Find the ratio of similitude
b) Find the lengths of EF and DE
c) Find the perimeter of DEF
6. If Pete is 6 foot tall and casts a shadow of 5 feet, while at the same time Alex casts a shadow of only 4.2
feet, how tall is Alex?
7.
.
a) Find the ratio of similitude.
b) Find all of the missing lengths.
8. For each pair of similar figures, state the ratio of similitude, then use it to find x.
a.
b.
c.
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
9. Find the ratio of the perimeters of two similar prisms if the ratio of their surface areas is 121:100. What
is the ratio of their volumes?
10. Find the ratio of the lengths of corresponding sides in two similar triangular prisms if the ratio of their
volumes is 216:343. What is the ratio of their surface areas?
11. If two similar, basic coffee mugs have heights of 3 inches and 4 inches, what is the ratio of their volumes?
12. Find the ratio of the areas of two similar prisms if the ratio of the volumes is 8:729.
Name: _____________________________________________________________
Period: ______
6-1
13. Find the ratio of the lengths of the sides of two similar triangles if the ratio of the areas is 25:196.
14. ∆ABC is similar to ∆DEF. The ratio of the lengths of
to 3:1?
(1)
(2)
15. Given ∆ABC ~ ∆DEF such that
(3)
Date: _________________________
to the length of
is 3:1. Which ratio is also equal
(4)
. Which statement is not true?
(1)
(3)
(2)
(4)
16. If the ratio of the volumes of two similar figures is 27:8, what is the ratio of the perimeters?
17. If the ratio of the areas of two similar figures is 36:169, what is the ratio of the areas?
18. If ∆ABC ~ ∆ZXY, m<A = 50, and m<C = 30, what is m<X?
(1) 30
(2) 50
(3) 80
(4) 100
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
19. The ratio of the areas of two rectangles is 4:121. Find the perimeter of the larger rectangle is the
perimeter of the smaller rectangle is 62 inches. Round to the nearest hundredth.
20. The ratio of the sides of two similar triangles is 5:6. If the area of the smaller triangle is 50 km2, what is
the area of the larger triangle?
21. The ratio of the surface areas of two similar cylinders is 144:100. Find the volume of the larger cylinder
to the nearest tenth if the volume of the smaller cylinder is 75 m3.
22. If the ratio of the perimeters of two cubes is 5:7, find the volume of the larger cube if the volume of the
smaller cube is 486 m3. Round to the nearest tenth.
Name: _____________________________________________________________
Date: _________________________
Period: ______
6-1
23. If the ratio of the volumes of two spheres is 343:729, find the area of the smaller sphere if the surface
area of the larger sphere is 198 cm2. Round to the nearest tenth.
24. Triangle RST is similar to triangle XYZ with RS = 3 inches and XY = 2 inches. If the area of triangle RST is
27 square inches, determine and state the area of triangle XYZ, in square inches.
25. The sides of a triangle are 8, 12, and 15. The longest side of a similar triangle is 18. What is the ratio of
the perimeter of the smaller triangle to the perimeter of the larger triangle?
(1) 2:3
(2) 4:9
(3) 5:6
(4) 25:36