Name: ___________________________________________________________________
Date: ___________
Applying similarity of triangles with area and perimeter
Geometry CC (Mod 2 – L4)
When 2 figures have sides that are in proportion and angles that are equal in measure, we call them _____________
Opening Exercise
Given DABC ~ DA'B'C' pictured to the right:
a. Find the lengths of the missing sides.
b. Find the perimeter of the triangles.
c. Find the area of the triangles.
------------------------------------------------------------------------------------------------------------------------------------------------------------d. What is the relationship between the sides of the triangles of ABC to the sides of A ' B ' C ' ?
e. What is the relationship between the perimeter of ABC to the perimeter of A ' B ' C ' ?
f.
What is the relationship between the area of ABC to the area of A ' B ' C ' ?
g. Make a hypothesis comparing these relationships.
------------------------------------------------------------------------------------------------------------------------------------------------------------Example 1: Let’s test the hypothesis we made in the Opening Exercise
Given the similar triangles pictured to the right, find:
a. Ratio of the sides.
b. Ratio of the perimeters.
c. Ratio of the areas.
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Example 2: Given the similar rectangles pictured to the right, find:
a. Ratio of the sides.
b. Ratio of the perimeters.
c. Ratio of the areas.
Think Deeper: If the similar figures were 3-dimensional, what do you think is the relationship between the volumes of
the figures?
PRACTICE EXERCISES:
1. Two triangles are similar. The sides of the smaller triangle are 6,4,8. If the shortest side of the larger triangle is
6, find the length of the longest side.
2. If MNPQ~XYZW, find the scale factor of MNPQ to XYZW and the perimeter of each polygon.
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3. The sides of a triangle are 8, 5, and 7. If the longest side of a similar triangle measures 24, find the perimeter
of the larger triangle.
4. On the blueprint of the apartment shown, the balcony measures 1 inch wide by 1.75 inches long. If the actual
length of the balcony is 7 feet, what is the perimeter of the balcony?
5. Find the ratio of the areas of two similar triangles in which the ratio of the pair of corresponding sides is 3:2.
6. Find the ratio of the lengths of a pair of corresponding sides in two similar polygons if the ratio of the areas is
4:25.
7. The sides of a triangle are 7, 8 and 10. What is the length of the shortest side of a similar triangle whose
perimeter is 75?
3. Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet above
the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the cliff
with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from the cliff,
to the nearest tenth, is Catarina’s boat?
8. Equilateral triangle MNP has a perimeter of 12𝑎 + 18𝑏. ̅̅̅̅
𝑄𝑅 is a midsegment. What is the length of QR?
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Name: ___________________________________________________________________
Applying similarity of triangles with area and perimeter-HOMEWORK
Date: ___________
Geometry CC (Mod 2 – L4)
1. Two triangles are similar. The sides of one triangle are 4,8 and 10. If the shortest side of the second triangle is
12, find the length of the missing sides.
2. Rectangle ABCD has a width of 8 yards and a length of 20 yards. Rectangle QRST, which is similar to rectangle
ABCD, has a length of 40 yards. Find the scale factor of rectangle ABCD to rectangle QRST and the perimeter
and area of each rectangle.
3. Two triangles are similar. The sides of the smaller triangle are 6, 4, and 8. The perimeter of the larger similar
triangle is 27. Find the length of the shortest side of the larger triangle.
4. Find the ratio of the areas of two similar triangles in which the ratio of the pair of corresponding sides is 5:3.
5. *Find the ratio of the volumes of two similar triangles in which the ratio of the pair of corresponding areas is
9:16.
6. Delroy’s sailboat has two sails that are similar triangles. The larger sail has sides of 10 feet, 24 feet, and 26
feet. If the shortest side of the smaller sail measures 6 feet, what is the perimeter of the smaller sail?
7. In the diagram of
and
,
. What is the length of
,
,
,
?
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8. In the diagram of
and
,
,
. What is the length of
,
,
?
9. Lines that appear parallel, are parallel. Find the value of x.
10. As shown in the diagram below,
,
,
,
, and
.
a) Find x.
b) What is the length of
?
11. The accompanying diagram shows two similar triangles. Find x.
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