Name: _________________________________
Special Right Triangles
Radical Form
Perfect Square Number Bank
144



64
36
25
16
4
In the bank, find a perfect square that is a factor of the number under the radical sign.
Simplify the radical.
Use a calculator to verify that you simplified correctly.
50 =

8 =

50 =

8 =

50 =
8 =
32 =

432 =

32 =

432 =

32 =
432 =
128 =

192 =

128 =

192 =

128 =
192 =
288 =

72 =

288 =

72 =

288 =
72 =
48 =
108 =
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Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Just Alike
Draw a line segment that bisects each of the following shapes into two congruent parts. As
appropriate, identify the bisecting segments as an altitude, a diameter, a diagonal, or a height.
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Supporting STAAR™ Achievement: Geometry
Special Right Triangles
Square Page
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Supporting STAAR™ Achievement: Geometry
Special Right Triangles
Triangle Page
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Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Special Triangles Part I
1. Cut out each of the squares from the Square Page.
2. Fold each square along a diagonal. Keep your square folded.
3. What do you know about the measures of the angles of the triangle formed by two sides of the
square and the diagonal?
4. Using your ruler, complete the table below. Measure to the nearest tenth of a centimeter.
Square
Length of one side
Length of the diagonal
large square
medium square
small square
5. Use the lengths of the legs of each triangle and the Pythagorean Theorem to calculate the
length of the hypotenuse of each triangle.
Length of the hypotenuse
Triangle
Pythagorean Theorem
Radical Form
(simplify)
Decimal Form
large triangle
medium triangle
small triangle
6. The lengths of each of the hypotenuses are multiples of _______.
7. The length of the hypotenuse is ______ times the length of one leg.
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Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Special Triangles Part II
1. Cut out each of the equilateral triangles from the triangle page. What do you know about the
angles of each triangle?
2. Fold each triangle along the altitude. Keep your triangle folded.
3. What do you know about the measures of the angles of the new triangle formed?
4. Using your ruler, complete the table below. Measure to the nearest tenth of a centimeter.
Triangle
Length of hypotenuse
Length of shorter leg
Length of longer leg
large triangle
medium triangle
small triangle
5. Use the lengths of the legs of each triangle and the Pythagorean Theorem to calculate the
length of the hypotenuse of each triangle.
Triangle
Pythagorean Theorem
Length of the longer leg
Radical Form
Decimal Form
large triangle
medium triangle
small triangle
6. The length of the longer legs are multiples of ______.
7. The length of the longer leg is ______ times the length of the shorter leg.
8. The hypotenuse is ______ the length of the shorter leg.
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Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Special Right Triangles Notes Page
Fill in each blank with the appropriate angle measure and side length from the table.
Angle Measure
45˚
45˚
90˚
Side Length
x
x
x 2
Use what you have learned about special right triangles to solve the following problems.
______ ft
2 ft
______ ft
Angle Measure
45˚
45˚
90˚
Side Length
x
x
x 2
Actual Length
____ miles
Angle Measure
45˚
45˚
90˚
Side Length
x
x
x 2
Actual Length
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____ miles
1.3 miles
Supporting STAAR™ Achievement: Geometry
Special Right Triangles
Fill in each blank with the appropriate angle measure and side length from the table.
Angle Measure
30˚
60˚
90˚
Side Length
x
x 3
2x
Use what you have learned about special right triangles to solve the following problems.
Angle Measure
Side Length
30˚
60˚
90˚
x
x 3
2x
4 ft
______ ft
Actual Length
______ ft
______ yds
9 yds
______ yds
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Angle Measure
30˚
60˚
90˚
Side Length
x
x 3
2x
Actual Length
Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Applying the Rules
For each word problem in the table, draw or complete the picture you would use to solve the
problem; then provide the solution to the word problem in the third column. Pictures and solutions
have been provided for some problems in the Starter Bank.
Starter Bank
20.78 in
8
8
12
27.31 ft
16.97 ft
8
Word Problem
Picture
Solution
A rabbit is on the ground and
looks up at a 60 angle to see
a hawk flying directly above
his rabbit hole. The rabbit is
12 feet from his hole.
Approximately how high is the
hawk above the rabbit’s hole?
The front of a tent is in the
shape of an equilateral triangle
with side lengths of 8 feet. A
tent pole is placed from the
peak of the tent to the center
of the opposite side (floor of
the tent). What is the
approximate height of the
pole?
8
Lindsey is decorating the top
of a square present with
ribbon. Each side of the square
is 12 inches. The ribbon will go
along both diagonals of the
top of the present.
Approximately how much
ribbon does she need?
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Supporting STAAR™ Achievement: Geometry
Special Right Triangles
Word Problem
A brace to hold a picture
frame is in the shape of an
isosceles right triangle. The
longest side of the brace
measures 8 feet. What is the
approximate length of each leg
of the brace?
Picture
Solution
8
Isabelle is creating a pattern
for her tile floors. She is using
three 30-60-90 triangles as
a part of the design. The
longer leg of one triangle is
24 inches. What is the
approximate total area of the
three-triangle design?
A board is leaning up against a
wall at a 45 angle. The base
of the board is 12 feet from
the wall. Approximately how
long is the board?
The diagonal of a rectangular
movie screen is 8 meters long
and forms a 30 angle with one
side of the screen. What is the
approximate area of the movie
screen?
A gardening co-op has decided
to place a fence around each
individual garden. Each garden
is in the shape of a



45 -45 -90 triangle. Each leg
of the triangle measures
8 feet. About how much
fencing is needed to enclose
one triangular garden?
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Supporting STAAR™ Achievement: Geometry
Special Right Triangles
Applying the Rules Cards*
33.94
8
a
a
60
16.97
60
4 ft
12 ft
20.78
24 in
8m
b
60

30
60
a
a
27.71
27.31
a
a
12
45
45
6.93
12 ft
5.66
a
8
a
8
45
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45
498.86
Supporting STAAR™ Achievement: Geometry
Name: _________________________________
Special Right Triangles
Evaluate: Special Right Triangles
1
Jenna is flying a kite on a very windy day. The kite string makes a 60 angle with the ground.
The kite is directly above the sandbox, which is 28 feet away from where Jenna is standing.
Approximately how much of the kite string is currently being used?
A 56 feet
B 48.5 feet
C 40 feet
D 14 feet
2
Nicole is creating a support in the shape of a right triangle. She has a 92 cm-long piece of
wood, which is to be used for the hypotenuse. The two legs of the triangular support are of
equal length. Approximately how many more centimeters of wood does Nicole need to complete
the support?
A 130 cm
B 184 cm
C 260 cm
D 276 cm
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Supporting STAAR™ Achievement: Geometry
Special Right Triangles
3
∆FGH is an equilateral triangle. Which value is closest to the perimeter of ∆FGJ?
G
22 in
F
H
J
A 39 in
B 52 in
C 62 in
D 66 in
4
Which of the following could be the side lengths of 45 -45 -90 triangle?
A 2 in, 4 in, 2 2 in
B 2 in, 4 in, 2 3 in
C 2 in, 2 in, 2 2 in
D 4 in, 4 in, 4 3 in
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Supporting STAAR™ Achievement: Geometry