Download Geometry Unit 3B SE - Clark Magnet High School

Glendale Unified School District
Math Curriculum
Geometry
Unit 3A: Right Triangle Trigonometry
Student Edition
Table of Contents
Lesson 1
Description
Geometric Mean and the Pythagorean Theorem
Lesson 2
Special Right Triangles
Lesson 3
Introduction to Basic Trigonometric Ratios
Standards
G.SRT.6
G.SRT 6
G.SRT 7
G.SRT.8
G.SRT.8.1
Page #
1-10
11-16
17-40
Pythagorean Theorem Proof
In previous classes and lessons we have used the Pythagorean Theorem.
Pythagorean Theorem: In a right triangle, the square of the
hypotenuse is equal to the sum of the squares of the other two sides
You probably know it as +
= . There are dozens of ways to prove it, and we are going to do one of them in this
worksheet. Let’s start by looking at a right triangle like the one below.
X
J
Y
A
In order to do this proof, we are going to use the properties of similar triangles. Use the space below to explain how you
know that
. Drawing the triangles separately may help.
--1--
Pythagorean Theorem Proof Page 2
Now that we know all of the triangles are similar we are going to look at some proportions. We have removed the
original letters and have put in values for lengths. a, b, and c have been put in their appropriate places.
x
c
a
y
b
Use the space below to draw the three similar triangles separately.
Look at your triangles above and use what you know about similar triangles to fill in the proportions below.
=
=
--2--
Pythagorean Theorem Proof Page 3
x
c
a
y
b
We are going to focus on two the of the proportions that you found on the previous page.
=
=
Use cross multiplication on the proportions above to find
=________
=________
Using your equations above combine them to find
+
= __________________
Use a little factoring magic in the space below to finally prove that
--3--
+
=
.
--4--
Lesson 1 Unit 3B
U
Pythagorean Theorem Worksheet
For each triangle find the missing length. Round your answer to the nearest tenth. Then find the
AREA and the PERIMETER.
1.
x
5
2.
13
19
x
17
x = _______
x = _______
Area = ______
Area = ______
Perimeter = _______
Perimeter = _______
3.
4.
x
5
14
20
10
x
x = _______
x = _______
Area = ______
Area = ______
Perimeter = _______
Perimeter = _______
For #5-9, c is the hypotenuse of the right triangle ABC with sides a, b & c.
5. a = 12, b = 5, c = _______
6. a = 8, b = ______, c = 10
7. a = 15, b = ______, c = 17
8. a = ______ , b = 40, c = 50
9. a = _______, b = 2, c = 4
--5--
Lesson 1 The Pythagorean Theorem Worksheet
10. Find a third side so that the three numbers form sides of a right triangle.
a)
______ , 9, 41
b) ______, 13, 85
11. Ms. Heart tells you that a right triangle has a hypotenuse of 13 cm and a leg of
5 cm. She asks you to find the other leg of the triangle. What is your answer?
12. Two joggers run 8 miles north and then 5 miles west. What is the shortest
distance, to the nearest tenth of a mile, they must travel to return to their
starting point?
13. Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet long
and the bottom of the house is 6 feet across. What is the height of his dog
house, in feet, at its tallest point?
14. To get from point A to point B you must avoid walking through a pond. To
avoid the pond, you must walk 34 meters south and 41 meters east. To the
nearest meter, how many meters would be saved if it were possible to walk
through the pond?
15. A suitcase measures 24 inches long and the diagonal is 30 inches long. How
much material is needed to cover one side of the suitcase?
--6--
Lesson 1 Unit 3B
U
Pythagorean Theorem Practice Problems HW
For each triangle find the missing length. When necessary, round your answer to the nearest tenth.
1.
2.
x
x
12
3.
3.
x
10
7 7
x
24
9
4.
x
5.
6
24
1
6.
6
1
x
10
x
7.
8.
21
x
8
9.
x
24
30
x
9
5
y
3
Find x and y.
10. A ladder is leaning against the side of a 10m house. If the base of the ladder
is 3m away from the house, how tall is the ladder? (Round to tenths place)
Draw a diagram and show all work.
--7--
Lesson 1 The Pythagorean Theorem Practice Problems - HW
11. What is the length of the diagonal?
40 yards
30 yards
12. What is the length of BD?
B
13 cm
A
C
D
10 cm
13. Use the Pythagorean Theorem to find out if these are right triangles. Justify
your answers.
a)
b)
c)
3cm
6.4m
4cm
5cm
5.6m
--8--
12.8 km
2.5m
16 km
9.6 km
Lesson 1 Geo. Mean Unit 3B G.SRT.5
NAME: _______________________________
1
1. Which of these proportions represent a geometric mean?
a) G.M. or Not G.M.
4 2
6 3
b) G.M. or Not G.M.
2 4
4 8
c) G.M. or Not G.M.
2
2 5
10
2 5
d) G.M. or Not G.M.
1
5
2 10
2. Determine the Geometric Mean of the two given numbers. (Exact Answers Only)
a)
4 and 9
b)
5 and 20
c)
1 and 25
d)
4 and 16
e)
8 and 9
f)
4 and 8
g)
2 and 75
h)
10 and 14
3. A geometry student says; “I got lost in that lesson - I wrote down that AB 2 AD AC but I have no idea
where it comes from.” Help this student - explain where AB 2 AD AC comes from.
B
o
A
x
D
4. Find the missing values. (If not a whole number, leave it in simplest radical form)
a)
b)
x
z
y
4
9
x = ________ y = _________
z = ________
c)
x
z
y
6
8
x = ________ y = _________
z = ________
--9--
x
z
y
14
2
x = ________ y = _________
z = ________
C
Lesson 1 Geometric Mean Unit 3B
G.SRT.5
2
5. Find the missing values. (If not a whole number, then round to two decimal places.)
a)
b)
c)
w
15
8
z
x
z
x
x
y
y
20
8
y
9
x = ________ y = _________
z = ________
d)
15
z
10
x = ________ y = _________
x = ________ y = _________
z = ________ w = _________
e)
f)
5
8
x
x
y
z
x = ________ y = _________
z = ________
z = ________
13
x
z
y
x = ________ y = _________
z = ________
--10--
12
5
z
y
x = ________ y = _________
z = ________
Lesson 2 Special Right Triangles
G.SRT.5
NAME: _______________________________
1
1. Solve for the missing information. (EXACT ANSWERS ONLY)
a)
b)
c)
d)
y
30°
6 3
x
y
7
45°
30°
y
y
60°
x
10
x
6 2
x
x = __________
y = __________
e)
x = __________
y = __________
f)
x = __________
y = __________
g)
x = __________
y = __________
h)
y
y
45°
60°
x
16 3
x
8 5
y
30°
45°
8 2
8
y
x
x
x = __________
y = __________
i)
x = __________
y = __________
j)
k)
y
x
x = __________
y = __________
l)
y
8 3
x = __________
y = __________
m)
n)
30°
x = __________
y = __________
x = __________
y = __________
p)
x
6 2
45°
30°
x
60°
x
60°
2
12 12
o)
y
x
5
y
x
x = __________
y = __________
12 6
y
x
18 2
45°
60°
x = __________
y = __________
45°
12 3
y
12.44
y
y
x
x = __________
y = __________
q)
x = __________
y = __________
r)
y
s)
11 2
2
x = __________
y = __________
x
y
8 2
30°
x = __________
y = __________
x = __________
y = __________
t)
x
x
45°
x = __________
y = __________
y
45°
7
y
10
x = __________
y = __________
--11--
60°
x
x = __________
y = __________
Special Right Triangles
G.SRT.5
2
2. Solve for the missing information. (EXACT ANSWERS ONLY)
a)
b)
c)
x
y
x
y
10
e)
9
f)
x = __________
y = __________
8 8
x
11
21
x
x
30°
y
i)
x = __________
y = __________
h)
45°
y
x = __________
y = __________
y
g)
3
14
y
x = __________
y = __________
x
j)
y
x = __________
y = __________
k)
x = __________
y = __________
l)
y
x
x
y
45°
30
30°
x
60°
x
x = __________
y = __________
15
y
y
6 2
x = __________
y = __________
45°
y
x = __________
y = __________
6
45°
30°
30°
x = __________
y = __________
x
5
x
45°
60°
d)
60°
x = __________
y = __________
x = __________
y = __________
3. Determine the area of the following. (EXACT ANSWERS ONLY)
a)
b)
c)
60°
9
d)
45°
6
45°
8 2
12
60°
Area = ___________
Area = ___________
Area = ___________
--12--
Area = ___________
Special Right Triangles
G.SRT.5
3
3. Solve for the missing information. (EXACT ANSWERS ONLY)
a)
b)
z
c)
z
4 2
60°
60°
45°
y
10 2
y
60°
45°
x
45°
x
8 3
y
z
x
x = __________
x = __________
x = __________
y = __________
y = __________
y = __________
z = __________
z = __________
z = __________
d)
e)
z
f)
z
8
30°
60°
y
30°
x
z
y
30°
60°
60°
x
x
9 3
y
3 3
x = __________
x = __________
x = __________
y = __________
y = __________
y = __________
z = __________
z = __________
z = __________
4. Determine the area of the following. (EXACT ANSWERS ONLY)
a)
b)
c)
6
12
60°
Area = ___________
10 6
10 3
Area = ___________
Area = ___________
f)
12 3
30°
30°
60°
15 3
Area = ___________
60°
60°
60°
Area = ___________
e)
d)
Area = ___________
--13--
60°
--14--
Name _____________________________________ Per _______ Date ______
Special Right Triangles and Geometric Mean Review Unit 3B
(Give all answers in simplified exact radical form)
1.
2.
3.
4 10
_____
8
13 2
_____
____
____
_____
_____
4.
5.
6.
7 3
30°
24
9
7.
8.
9.
_____
18
____
7
3 6
10.
11.
12. Equilateral Triangle
J
15
_____
16
____
28
_____
--15--
R
_____
L
Q
13. The length of the diagonal of a square is 12 inches. Find the length of one side of the square.
14 The length of one side of an equilateral triangle is 6 3 meters. Find the length of the altitude of the triangle.
15. The length of the altitude of an equilateral triangle is 12 feet. Find the length of one side of the equilateral triangle.
What is the perimeter of the equilateral triangle?
16. The perimeter of an equilateral triangle is 39 cm. Find the length of the altitude of the triangle.
17. The length of the diagonal of a square is 18 mm. Find the perimeter of the square.
18. The diagonal of a rectangle is 12 in and intersect at an angle to make a 60° angle. Find the perimeter of the
rectangle.
Geometric Mean (Give exact answers)
19.
20. Find x
21. Find x
50
40
x
4
6
x
22. Find x
21
x
4
23.
24.
x
30
x
3
9
9
7
--16--
x
6
--17--
--18--
--19--
--20--
Trigonometric Ratios
G.SRT.6
NAME: _______________________________
The Sine Ratio (sin)
Opposite
sin
Hypotenuse
The Cosine Ratio (cos)
Adjacent
cos
Hypotenuse
1
The Tangent Ratio (tan)
Opposite
tan
Adjacent
1. Match the following.
C
a) _______ Opposite Leg to A
b) _______ Sine Ratio of C
c) _______ Opposite Angle to AB
B
A
d) _______ The Hypotenuse
1. A
e) _______ Adjacent Leg to A
f) _______ Tangent Ratio of C
BC
g) _______ Reference angle if
is the Cosine Ratio.
AC
h) _______ Adjacent Leg to C
2.
B
3.
C
4. AB
i) ________ Cosine Ratio of A
7.
BC
AC
8.
AB
AC
9.
BC
AB
5. BC
j) ________ The Longest Side
BC
k) _______ Reference angle if
is the Sine Ratio.
AC
10.
6. AC
AB
BC
2. Label the sides based of the triangle using the reference angle -- (O) for Opposite, (A) for Adjacent and
(H) for Hypotenuse. After you have labeled the triangle, then choose which trigonometric ratio that you
would use to solve for the missing info.
a)
b)
c)
C
12 cm
29 cm
B
30°
A
A
B
x
d)
A
x
C
C
13°
B
21 cm
25 cm
C
34 cm
B
A
SIN
COS
TAN
e)
SIN
COS
TAN
f)
C
B
x
66°
SIN
COS
g)
A
x
18 cm
67°
12 cm
A
COS
TAN
SIN
COS
SIN
COS
TAN
h)
C
8 cm
B
B
35°
C
x
x
B
SIN
TAN
25 cm
TAN
SIN
--21--
COS
C
A
TAN
63°
34 cm
SIN
COS
A
TAN
Trigonometric Ratios
G.SRT.6
2
3. Solve the angle. (Round all final answers to 2 decimals places)
a)
b)
c)
d)
11 cm
12 cm
21 cm
6 cm
11 cm
24 cm
15 cm
16 cm
____________
____________
____________
4. Solve for the side x. (Round all final answers to 2 decimals places)
a)
b)
c)
d)
x
50°
x
x
28°
11 cm
21 cm
8 cm
53°
41°
30 cm
x
x ____________
5. Solve for the side x. (Round all final answers to 2 decimals places)
a)
b)
x
c)
d)
x
62°
18 cm
24 cm
x = ____________
34°
x
12 cm
48°
x = ____________
x = ____________
--22--
6 cm
33°
x
x = ____________
Trigonometric Ratios
G.SRT.6
3
6. Solve for the missing information. (Round all final answers to 2 decimals places)
a)
b)
c)
d)
23°
x
33°
x
11 cm
15 cm
34 cm
8 cm
23 cm
16 cm
= ____________
e)
= ____________
f)
g)
h)
2 cm
30°
5 cm
x
13 cm
x
8 cm
46°
15 cm
= ____________
i)
____________
j)
k)
15 cm
67°
38°
x
= ____________
l)
x
18 cm
6 cm
4 cm
7 cm
13 cm
42°
x
= ____________
m)
n)
x
15°
22 cm
o)
p)
6 cm
11 cm
45°
57°
x
15 2 cm
15 cm
x
= ____________
--23--
--24--
Understanding Trigonometric Ratios
G.SRT.6
NAME: _______________________________
1
1. Earlier we learned about two very special triangles, the 30 -60 -90 and the 45 -45 -90 . Use the side
relationships that we learned to determine the exact values for sine, cosine and tangent for the angles of
30 , 60 and 45 . (Why didn’t I have to provide any measurements for the sides…..?)
sin 30
cos 30
60°
tan 30
30°
sin 60
cos 60
60°
tan 60
30°
sin 45
cos 45
tan 45
45°
2. A teacher asks (while looking at the trigonometry table), “What does the 0.6691 mean?” How would you
respond to that?
Degrees
42
Sine
0.6691
Cosine
0.7431
Tangent
0.9004
3. Thomas sees two triangles on the board in geometry class. From those he makes two claims:
CD EF
. How could he know this without having any
CB EG
numbers on the sides?
B
#1) that
C
24°
F
24°
D
G
CD
0.9135, how could he
CB
do this without any numbers on the sides? What did he type into his calculator to get this value?
#2) and that after clicking a couple of buttons on his calculator he states that
--25--
E
Understanding Trigonometric Ratios
G.SRT.6
2
4. Sarah, the student who sits next to you in geometry class, notices that the values for sine and cosine are
only from 0 to 1 for the angles 0 to 90 . She leans over and asks, “Why can’t sine and cosine be greater
than 1?” How would you respond to her?
5. A student who did very well in Algebra 1 looked at this trigonometry problem, and said “What a minute,
Tangent is the same as slope!!” Why would she says this? How is tangent the same as slope?
14 cm
38 cm
6. At 45 , the tangent value = 1.
a) What does that mean about the opposite and adjacent sides?
b) A student notices that angles greater than 45 produce a ratio greater than 1. What does this
mean about the opposite and adjacent sides?
6. Jessy claims that AB is the opposite side. Is he correct? Explain.
C
o
A
7. What does similarity have to do with Trigonometry?
--26--
B
The Relationship between Sine and Cosine G.SRT.7
NAME_______________________________
1
1. When looking closely at the trigonometry table a student notices that certain sine values are the same as
certain cosine values (partial table shown). What do the angles that have the same value have in common?
Degree
10
9
8
7
6
5
4
3
2
1
0
Sine
0.1736
0.1564
0.1392
0.1219
0.1045
0.0872
0.0698
0.0523
0.0349
0.0175
0.0000
Degree
80
81
82
83
84
85
86
87
88
89
90
Cosine
0.1736
0.1564
0.1392
0.1219
0.1045
0.0872
0.0698
0.0523
0.0349
0.0175
0.0000
–
3. Solve the following.
a) sin 42
b) cos 12
c) sin 45
e) cos 65
f) sin 78.5
4. Solve for the unknown.
a) sin (x – 5
3
d) sin ( x
4
)
1
x)
4
b) sin (2x – 17
e) sin (5x – 22
--27--
–4 )
(x –
c)
)
3
f) sin ( x 3
4
)
Trigonometric Ratios
G.SRT.6
2
Geometry Unit 3B - Trigonometric Ratios of Acute Angles
Triangle ABC is a right triangle with sides of lengths a, b, c, and right angle at C. Find the
unknown side length using the Pythagorean Theorem, and then find the values of the three
trigonometric ratios for angles A and B. Give exact values where needed.
1. a = 5, b = 12
sinA=
sinB=
B
2. a = 3, b = 4
a = 3, b = 8
cosA =
cosB=
tanA=
tanB=
sinA=
sinB=
cosA =
cosB=
tanA=
tanB=
sinA=
sinB=
cosA =
cosB=
tanA=
tanB=
C
A
B
C
A
B
C
Use the given trigonometric ratio to find exact values for sinA, cosA, or tanA.
3. sin A =
cos A =
tan A =
4. cos A =
sin A =
tan A =
5. tan A = 1
sin A =
cos A =
6. cos A =
sin A =
tan A =
--28--
A
Angles of Elevation and Depressions
G.SRT.8
NAME: _______________________________
1. Choose the correct angle number for the provided description.
5
10
11
6
7
18
4
13
8
22
21
19
2
3
1
12
20
9
9
16
15
17
14
a) the angle of elevation from the CAR to the top of the DINER is _________.
b) the angle of depression from the top of the TALL BUILDING to the DINER is _________.
c) the angle of elevation from the PLANE to the HELICOPTER is _________.
d) the angle of depression from the top of the DINER to the BOY is ________.
e) the angle of depression from the HELICOPTER to the PLANE is ________.
f) the angle of depression from the PLANE to the top of the DINER is __________.
g) the angle of elevation from the BOY to the top of the DINER is __________.
h) the angle of depression from the top of the TALL BUILDING to the top of the CAR is _____.
i) the angle of depression from the HELICOPTER to the top of the TALL BUILDING is ______.
j) the angle of elevation from the top of the DINER to the top of the TALL BUILDING is ______.
k) the angle of elevation from the top of the DINER to the PLANE is _______.
l) the angle of depression from the top of the DINER to the CAR is _________.
m) the angle of elevation from the BOY to the front of the PLANE is _________.
n) the angle of depression from the front of the PLANE to the BOY is _________.
o) the angle of elevation from the TALL BUILDING to the HELICOPTER is _________.
--29--
1
Angles of Elevation or Depressions
G.SRT.8
2
2. Circle (or Draw) the side or angle that is represented by the description.
a) The Leaning Ladder
Height on the wall that
the ladder reaches.
b) The Leaning Ladder
c) The Leaning Ladder
The distance from
the foot of the ladder
to the wall.
e) Flying a Kite
The angle the ladder
forms with the wall.
d) The Shadow
The length of his shadow.
f) Flying a Kite
The length of the string.
The height of the kite.
What are some of the assumptions that are made about the kite example so that it works easily as a
trigonometry question?
g) The Support Guy Wire
h) The Support Guy Wire
The distance from the
base of the tree to where
the guy wire is fastened
to the ground.
The angle between the
antenna and the guy
wire.
i) The Support Guy Wire
The height of where the
guy wire is fastened
to the antenna.
j) The Support Guy Wire
The angle formed
between the wire
and the ground.
What are some of the assumptions that are made about the guy wire example so that it works easily as a
trigonometry question?
--30--
Angles of Elevation or Depressions
G.SRT.8
3
3. Create the diagram for the following descriptions. Label the diagram completely including putting the x for
the unknown missing value.
a) A young boy lets out 30 ft of string on his kite. If
the angle of elevation from the boy to his kite is 27°,
how high is the kite?
DIAGRAM
b) A 20 ft ladder leans against a wall so that it can
reach a window 18 ft off the ground. What is the
angle formed at the foot of the ladder?
DIAGRAM
c) To support a young tree, Jack attaches a guy wire
from the ground to the tree. The wire is attached to
the tree 4 ft above the ground. If the angle formed
between the wire and the tree is 70°, what is the
length of the wire?
DIAGRAM
d) A helicopter is directly over a landing pad. If Billy is
110 ft from the landing pad, and looks up to see the
helicopter at 65° to see it. How high is the
helicopter?
e) A man casts a 3 ft long shadow. If the sun’s rays
strike the ground 62°, what is the height of the man?
f) A man in a lighthouse tower that is 30 ft. He spots a
ship at sea at an angle of depression of 10°. How far
is the ship from the base of the lighthouse?
DIAGRAM
DIAGRAM
4. Now solve them.
a)
DIAGRAM
b)
c)
x = _______________
d)
x = _______________
e)
x = _______________
f)
x = _______________
--31--
x = _______________
--32--
Using Trig. Ratios to Solve Problems
G.SRT.8
NAME: _______________________________
1. Solve the following problems. (All answers to 2 decimals places, unless otherwise instructed.)
a) A tree casts a shadow 21 m long. The angle of elevation of the sun is
55 . What is the height of the tree?
b) A helicopter is hovering over a landing pad 100 m from where you are
standing. The helicopter’s angle of elevation with the ground is 15 .
What is the altitude of the helicopter?
c) You are flying a kite and have let out 30 ft of string but it got caught in
a 8 ft tree. What is the angle of elevation to the location of the kite?
d) A 15 m pole is leaning against a wall. The foot of the pole is 10 m from
the wall. Find the angle that the pole makes with the ground.
e) A guy wire reaches from the top of a 120 m television transmitter
tower to the ground. The wire makes a 68 angle with the ground. Find
the length of the guy wire.
f) An airplane climbs at an angle of 16 with the ground. Find the ground
distance the plane travels as it moves 2500 m through the air.
--33--
1
Using Trigonometric Ratios to Solve Problems
G.SRT.8
2
g) A lighthouse operator sights a sailboat at an angle of depression of 12 .
If the sailboat is 80 m away, how tall is the lighthouse?
Solve the following problems.
2. a) How long is the guy wire?
b) What is the angle formed
between the guy wire and the
ground?
3.a) What is the length of the line
of sight from the man to the
helicopter?
b) What is the angle of elevation
from the man to the helicopter?
4.a) A field has a length of 12 m
and a diagonal of 13 m. What is
the width?
b) What is the angle formed
between the diagonal and the
width of the field?
13 m
12 m
5.a) A 5 ft 11 inch women casts 3 ft shadow. What is the angle that the
sun’s rays make with the ground?
6. a) A ramp is 18 m long. If the
horizontal distance of the ramp is
17m, what is the vertical distance?
b) What is the angle of elevation of
the ramp?
--34--
Using Trigonometric Ratios to Solve Problems
G.SRT.8
3
7. a) Using the drawbridge diagram, determine the distance from one side to the other. (exact answer)
45°
40 ft
45°
40 ft
b) Now that you know the distance from side to side, determine how high the drawbridge would be if the
angle of elevation was 60 .(exact answer)
60°
60°
40 ft
c) How far apart would the drawbridge be if the angle of elevation of the drawbridge was 20 ?
x
20°
--35--
20°
--36--
Solving Problems using Trig. G.SRT.8
NAME: _______________________________
1
For each problem, first draw the diagram and then solve for the requested information.
(All answers to 2 decimals places, unless otherwise instructed.)
1. Sharon is flying a kite on a string 130 m long. Determine the height of the kite if the string is at an angle of
37 to the ground.
2. An airplane is flying at an altitude of 6000 m over the ocean directly toward
an island. When the angle of depression of the coastline from the airplane is
14 , how much farther does the airplane have to fly before it crosses the
coast?
3. A loading ramp is 25 m long with a height of 10 m. What is the horizontal
distance of the ramp and what is the angle of incline that the ramp forms with
the ground?
4. A telephone pole casts a shadow 18 m long when the sun’s rays strike the
ground at an angle of 70 . How tall is the pole?
5. How long must a brace to a Satellite Dish be if it is attached to the antenna
3 ft above the ground and forms an angle of 68 with the antenna?
6. Mike Patterson looks out the attic window of his home, which is 22 ft
above the ground. At an angle of elevation of 35 he sees a bird sitting at the
very top of the large high rise apartment building down the street. How tall is
the high rise apartment building, if the two buildings are 75 ft apart?
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