Download 5.2 Special Right Triangles

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5.1
Radical Review & Pythagorean Theorem
Radical Review
SIMPLIFY FATTY RADICALS
MULTIPLYING RADICALS
1. Make a factor tree
2. Circles all the prime #’s
3. Square Couples go out (as one) & Single people stay home
1. multiply the “like terms”.
(outside x outside; inside x inside)
a. √8
d. √3 ∙ √2
c. 4√125
b. √108
e. 3√5 ∙ 7√5
2
f. (9√2)
Example 1: Simplify.
1.
√42
√24
2.
5. 5√3 ∙ 7√3
4. √5 ∙ √2
6. 6√3 ∙ 8√7
2
2
2
8. (4√3)
7. (10√2)
√48
3.
9. (7√2)
SOLVING RADICAL EQUATIONS (hint: √𝑛 ∙ √𝑛 = 𝑛)
1.
𝑥√5 = 25
2.
√7
2
6
3.
=𝑎
1
𝑦√3 = 18
4.
8
𝑚
=
√2
3
Right triangle
HYP: _______
K
Hypotenuse:
LEGS:_______ & ______
Legs:
I
M
Pythagorean Theorem
___
-___
In a right triangle, if a and b are legs and c is the hypotenuse,
then 𝑎2 + 𝑏 2 = 𝑐 2
___
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
___
three whole numbers that make up the sides of a right triangle.
(Keep your eyes open for them!)
*Pythagorean Triple:
Example 2: Leave you answers as a simplified radical.
1. Solve for 𝑥.
2. Solve for 𝑥.
3. Find the length of HP.
3. Solve for x.
A
P
x
2√6
H
4. Find the values of the variables.
x
5
8
4
y
6
2
Example 3: Using the Pythagorean Theorem
1. A 20 ft ladder is leaning against the side of a house. The
distance from the foot of the ladder to the bottom of the
house is 12 ft. How high up the wall is the ladder
2. How much shorter is it to get from your house to the
McDonald’s if you don’t stop at Maria’s?
Maria's House
16 mi
30 mi
Your House
3. If a tree breaks as shown, what is the original height of
the tree? Round your answer tot eh nearest tenth.
4. If you start at your house, walk to the basketball courts,
then your friends house, and finally back to your house,
how far did you walk?
B-Ball courts
1500 yards
Your house
15 ft
800 yards
22 ft
Friend's house
CONVERSE OF THE PYTHAGOREAN THEOREM
Can be used to check if a figure is a right triangle.
If three sides of a triangle satisfy 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 , then the three sides form a right triangle.
Example 4: Use the converse.
1. Is the triangle below a right
triangle?
2. Is the triangle below a right triangle?
How do you know which
side will be 𝑐?
3. Can the following side lengths form a
right triangle?
A. 10, 6, 8
B. 8, 3√7, 12
3
5.2
Special Right Triangles
Investigating Special Right Triangles (30°-60°-90°)
30°
Use the triangles
below to answer the following questions.
N
y
x
A
a. Are the triangles similar? If so, show work to prove it and write a
b. similarity statement.
5
8
x
4
30°
5
6
4
b. If the triangles are similar, what do we know about the sides?
860°
B
C
6
y
L
S
c. If NS = 4, AC = 2, and BC = 1, find LS.
D
30°
d. Use the Pythagorean theorem to find ̅̅̅̅
𝑁𝐿 and ̅̅̅̅
𝐴𝐵.
c
a
e. ∆𝐷𝐸𝐹 is also similar to ∆𝑁𝐿𝑆 and ∆𝐴𝐵𝐶 . Find a and c in ∆𝐷𝐸𝐹.
E
F
3
Example 1: Discovering patterns in a 30°-60°-90° ∆.
Label the sides of all three triangles based on the information gained above. Examine them
carefully. Do you notice a pattern?
N
A
60°
c
Describe your pattern below
and apply to the last triangle.
D
b
30°
30°
a
30°
30°
5
A
60°
L
B
C
E
5
6
30°- 60°- 90° TRIANGLE THEOREM
B
60°
c
b
In a 30-60-90 triangle, the extended
30°
60°
6
F
10
5
8
4
ratio of the sides is 1: √3 : 2
A
60°
60°
S
8
4
4
60°
5
8
4
6
C
30° - 60° - 90°
1
:
√3
:
B
2
a
Example 2: Find the value of the variable. If necessary, leave answers as simplified radicals.
4
60°
C
1. Solve for 𝑎.
2. Solve for 𝑥 and 𝑦.
3. If 𝑧 = 4√3, find the value of 𝑥.
𝑧
6
Example 3: Investigating Special Right Triangles (45°-45°-90°). Complete the following
investigation with your group. Record and apply your findings.
Use the similar triangles below to find…
1st: The missing leg and 2nd: The hypotnuse of each one (split class in 3 groups for this one)
R
X
D
X
45°
45°
y
x
45°
6
Z
45°
Z
Y
3
F
Example 4:DDiscovering patterns in a 45°-45°-90° ∆.
D
Label the sides of all three triangles based on the information gained above. Examine them
carefully. Do you notice a pattern?
6
45°
45°
Z
T
Y
45°
x
D
In a 45-45-90 triangle,
the extended ratio
of the
y sides is 1: 1: _______
y
x __________________
* Note: Legs are always
45°
F
S
45°-45°-90° TRIANGLE THEOREM
D
45°
F
E
6
6
45°
Describe your pattern below
and apply to the last triangle.
D
R
E
E
S
S
45° F
45°
4
45°
T
X
Y
3
D
c
a
x
7
45° - 45° - 90°
45°
D
45°
E
y
1
45°
x
:
1
:
√2
y
b
F
4
E
F
E
F
E
4
Example 5: Find the value of the4variables. Leave answers
as simplifiedF radicals.
4
5
E
1. Solve for 𝑥.
3. What is the value of x in the triangle?
2. If z = 9√2, then find the value of x.
Is this a
45-45-90 ∆?
Why?
x
14
10
𝑧
14
22
14
Identify the Special Right Triangle: Is it a 30°-60°-90°, 45°-45°-90°, or neither?
1. A
14
2.
C
3.
D
14
D
14
E
6√2
22
3
F22
11√3
I
B
E
F
MORE PRACTICE: Solve for the value of the variables.
G
1.
I
G
2.
H
3.
4.
6
H
5.
6.
5.3
Advanced Special Right Triangles
RECTANGLE
SQUARE
EQUILATERAL TRIANGLE
A diagonal divides a rectangle into
two _________________ ∆s.
A diagonal divides square into two
____________ ∆s.
The altitude (height) divides it into two
________________ ∆s.
Use Pythagorean Theorem
Use the ratio _______________
 Use the ratio _______________
Example 1
2. ΔHAT is an equilateral triangle, find the length of its
altitude (height).
1. YELP is a square and its diagonal has a length 8√2
inches. Find the length of one of its sides.
Y
E
P
L
A
H
3. Find the perimeter of the rectangle below
10
T
4. Find the side length of an equilateral triangle whose
height 21√3 meters. (Draw a picture!)
8
10
7
5. Find length of the diagonal of a square with whose
perimeter is 20 feet. (Draw a picture!)
6. If the altitude (height) of an equilateral triangle if 7√3,
find the perimeter of the triangle (Draw a picture!).
7. If the area of a square is 36 ft2, then what is the length of
its diagonal?
8. Find w and v.
v
w
60°
12
9. You are standing 18 feet from a building. The vertical
10. A symmetrical canyon is 4850 feet deep. A river runs
through the canyon at its deepest point. The angle of
depression from each side of the canyon to the river is
60°. Find the distance across the canyon. Round to the
nearest tenth.
distance from the ground to your eye is 5.5 feet. Determine
the height of the building. Round your answer to the nearest
tenth.
30°
Preview to 5.4 – Introduction to Trigonometric Ratios
Reference angle
Hypotenuse
Opposite
8
Adjacent
1. Using ∠J as your reference angle, label the
hypotenuse, opposite, and adjacent sides.
2. Would your answers change if your reference
angle was ∠L? Why or why not?
J
3. Can you use the right angle as your reference
angle? Why or why not?
L
K
Practice: Label an O for opposite side, A for adjacent side, and an H for hypotenuse based on the given angle.
1. Label from ∠𝐺
2. Label from ∠𝐾
3. Label from ∠𝑅
R
I
U
M
R
K
G
G
I
*Which side is always labeled the same regardless of what angle you start from? O, A, or H? Why?
5.4
Introduction to Trigonometric Ratios
Using similar triangles to understand tangent.
1. Which theorem or postulate can be used to prove JKL similar
to SRT? Explain.
45°
c
a
S
J
31°
R
10
3
K
45°
T
b
2. Find the mS. Explain.
5
6
L
3. Above you showed that the ratios of the corresponding sides of similar triangles are congruent. Now let’s exam the ratio
of the sides within each triangle. Fill in the value for the ratios below, as a fraction and a decimal:
∆𝑱𝑲𝑳
𝐾𝐿
𝐽𝐾
=
∆𝑺𝑹𝑻
=
and
𝑅𝑇
𝑆𝑅
=
=
4. What do you notice about the ratios?
5. KL and RT are opposite of J and S, respectively. JK and RT are adjacent to J and S, respectively. In any triangle
opposite
3
similar to JKL and SRT , using the reference angle with a measurement of 31° the ratio of the adjacent sides is 5 , or
0.6 as a decimal.
This ratio trigonometric ratio of
𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞
𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭
sides that you used above is referred to as tangent.
The SINE and COSINE ratio
9
Two other trigonometric ratios are sine =
𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞
and cosine =
𝐡𝐲𝐩𝐭𝐨𝐞𝐧𝐮𝐬𝐞
𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞
.
45°
c
1. Find the length of the hypotenuse of each triangle shown.
a
S
J
10
31°
K
45°
b
6
2. Using ∆𝐽𝐾𝐿 and ∆𝑆𝑅𝑇, find the sine and cosine ratios.
L
∆𝑱𝑲𝑳
45°
𝐾𝐿
opposite
Sine ∠𝑺= hyptoenuse
a
𝐽𝐿
∆𝑺𝑹𝑻
S
=
c =
=
and
31°
𝑅𝑇
𝑆𝑇
= 10
5
R
=
3
K
45°
adjacent
𝐽𝐾
𝐽𝐿
Cosine∠𝑺= hyptoenuse
= b =
=
𝑆𝑅
𝑆𝑇
and
T
=
=
6
Here’s How to MEMORIZE the TRIG RATIOS
SOH
CAH
TOA
cos ∠R =
sin ∠R =
S
tan ∠R =
a. 𝑠𝑖𝑛∠𝑆 =
=
=
d. 𝑐𝑜𝑠∠𝑆 =
=
=
b. 𝑠𝑖𝑛∠𝑅 =
=
=
e. 𝑐𝑜𝑠∠𝑅 =
=
=
58
42
T
R
40
c. What do you notice about 𝑠𝑖𝑛∠𝑆
and 𝑠𝑖𝑛∠𝑅?
f. What do you notice about 𝑠𝑖𝑛∠𝑆
and 𝑐𝑜𝑠∠𝑅?
Example 1
1. Find the trigonometric ratios of each right triangle.
B
5
C
12
𝑡𝑎𝑛∠𝐵 =
2. Find the trigonometric ratios of each right triangle.
J
𝑐𝑜𝑠∠𝐽 =
16
8√3
A
𝑡𝑎𝑛∠𝐽 =
𝑐𝑜𝑠∠𝐵 =
K
10
8
L
6
8
3. Find the sinW and tanW if cosW = 10.
4. Find the cosG and sinG if tanG = 15.
P
Y
I
G
W
G
13
6. Find sinV.
5. If cos 𝐴 = 85, then what is sin A and tan A?
M
30°
a. sin 𝐴 =
b. sin 𝐴 =
13
;
85
tan 𝐴 =
tan 𝐴 =
13
60°
28
17
A
W
13
84
13
34°
c. sin 𝐴 = 84; cos 𝐴 =
85
84
60°
84
d. sin 𝐴 = 85; cos 𝐴 = 13
O
H
14 3
A
82
28
V
84
85
60°
84
;
85
15
U
N
10
x
60°
x
10
EXTRA REVIEW
45
1. Do the following side lengths
form a right triangle?
15
11, 14, 5√3
2. Solve for 𝑚 and 𝑛.
5 13
28
34°
j
3. Solve for 𝑥 in the square shown below.
82
F
4. Find tanA and cos A.
G
x
h
10
I
a) Tan A =
20
15
E
G
H
11
5 13
b) Cos A =
5. Solve for 𝑥. If necessary, leave your answer as a
simplified radical.
6. Find the area of the equilateral triangle shown below.
𝑏ℎ
(𝐴∆ = 2 )
7. An investor owns a triangular plot of land as shown in the diagram.
a. Find the perimeter of the plot of land.
b. One acre of land is equivalent to 43,560 square feet. How many acres are
in this plot of land? Round to two decimal places.
12
5.5
Use Trigonometry to Find Missing Side Lengths
I. Choosing a Trigonometric Ratio
1st: Label the sides opposite, adjacent, and hypotenuse based on the reference angle
2nd: If OPPOSITE/HYPOTENUSE → choose SINE
If ADJACENT/HYPOTENUSE → choose COSINE
If OPPOSITE/ADJACENT → choose TANGENT
Example#1: Determine which trig ratio you will use based on the given side and the reference angle given.
a. Reference Angle: ∠𝐷
b. Reference Angle: ∠𝑇
c. Reference Angle: ∠𝐼
F
M
33°
5.991
5
G
T
11
H
19.416
V
6
46°
D
4.828
72°
I
E
• With a calculator, evaluate each trigonometric ratio using the angle measure.
a.
b.
c.
II. Use Trigonometric Ratios to Find Sides
HOW TO FIND MISSING SIDES OF RIGHT TRIANGLES
1. Mark your reference angle
2. Label the Hypotenuse, the Opposite and Adjacent side
(*based on given acute angle)
3. Circle the two that you are going to use
- what you WANT: the side you are solving for (the variable)
- what you HAVE: a side with a known value
14
x
35°
4. Set up your trig ratio (SOH-CAH-TOA) and solve
Example 2: Use Trigonometric Ratios to Find Side Lengths
1. Find the value of x. Round to the nearest tenth.
2. Find the value of x. Round to the nearest tenth.
13
3.(a) Find the lengths of both missing sides. Round to the nearest tenth.
(b) Is there another method you could have used to find the last missing side?
III. Three Methods to Find Missing SIDES
GIVEN 2 SIDES?
GIVEN 1 SIDE AND 1 ANGLE?
• Use the Pythagorean Theorem
• If it’s as SPECIAL right ∆?
Ex:
𝑎2 + 𝑏 2 = 𝑐 2
15
If it’s NOT a special one…
• Use a Trigonometric Ratio
Ex:
45°-45°-90°  1: 1: √2
63º
30°-60°-90° 1: √3: 2
11
11
SOH – CAH – TOA!
Example 3: Leave your answers as a simplified radical or round to the nearest tenth.
1. Find the length of ̅̅̅̅
𝐺𝐻.
G
2. Find x.
I
42
15
x
9
70°
H
3. Find x.
4. If b = 14 in the right triangle below, then what is the
value of a?
x
6
c
60°
b
22°
a
g
14
x
IV. Multiple-Choice Problems
1. In triangle 𝐷𝐸𝐹, which equation can be used to find the
D
value of 𝑥?
A. 𝑥 = 14 cos 61°
H
61°
14
6
A
A. 𝑥 = 6 tan 27°
14
6
B. 𝑥 = cos 61°
F
C. 𝑥 = 14 sin 61 °
D. 𝑥 =
2. In triangle 𝐻𝐴𝑀, which equation can be used to find the
value of 𝑥?
E
x
14
cos 61°
B. 𝑥 = tan 27°
27°
x
C. 𝑥 = 6 sin 27°
D. 𝑥 =
6
sin 27°
M
61°
14
x
3. In rhombus GERM , which equation can be used to find
E
̅̅̅̅? Explain your reasoning.
the length of 𝐸𝑌
G
A. 𝑥 = 16 sin 22°
6
B. 𝑥 =
16
sin 22°
16
Y
C. 𝑥 = 16 cos 22 °
27°
x
D. 𝑥 =
16
cos 22°
22°
R
4. Given rhombus 𝑀𝑁𝑂𝑃, if 𝑀𝑂 = 12 then find the
perimeter of rhombus 𝑀𝑁𝑂𝑃.
N
A. 8√3
B. 16√3
C. 16
D. 48√3
M
M
30°
S
O
P
5. To an observer on a cliff 360 m above sea level, the angle of depression to a ship is 28°. What is the horizontal distance
between the ship and the base of the cliff?
A. 360 sin 28° meters
360
B. sin 28° meters
C. 360 tan 28° meters
D.
15
360
tan 28°
meters
5.6
Cofunction Identities
In every right triangle there are two complementary angles.
M
• ∠_____ and ∠_____ are complementary because their sum in ______°.
• If 𝑚∠𝑀 = 53°, then find 𝑚∠𝑋.
X
A
• Use a calculator to find the sine and cosine value for ∠𝑀 and ∠𝑋.
(Round to the nearest hundredths)
• What do you notice about the values? Why should this make sense (use the diagram to explain).
*This pattern is called COFUNCTIONAL.
Cofunctional Identities: Cofunctions of complementary angles are equal.
Examples:
1. If sin 32° ≈ 0.53 and 𝑐𝑜𝑠𝐵 ≈ 0.53, then find the
measure of ∠𝐵.
2. Complete the statement below to make it true.
cos 46° = sin ______°
3. Write the trigonometric ratio that has the same value as
the given trigonometric ratio.
4. If tan 𝐿 =
S
T
a. cos 𝑆 =
20
,
21
then find sin 𝑉 and cos 𝐿.
V
b. sin 𝑇 =
L
c. cos 𝑇 =
T
V
5. Challenge Question: (Draw a sketch to help)
Given triangle 𝐴𝐵𝐶 is a right triangle where ∠𝐶 is the right angle. If tan 𝐵 = 1, ind the measure of ∠𝐵.
We also learned that cos𝐴 = sin ______. Write the sin 𝐴 in fraction form.
16
Review:
1. Which of the expressions below are equivalent to the value of 𝑥? Choose All That Apply.
O
36°
B
10
x
M
A. 𝑥 =
10
cos 54°
B. 𝑥 = 10 sin 36°
C. 𝑥 = 10 cos 54°
D. 𝑥 =
10
sin 36°
2. Figure 𝐺𝐻𝐼𝐾𝐽 is comprised of a rectangle and an equilateral triangle.
If the height of equilateral triangle 𝐽𝐼𝐾 is 24√3, then find the perimeter of 𝐺𝐻𝐼𝐾𝐽.
3. (a) Which angle are you NOT allowed to use as the reference angle?
B
(b) Which angle should you use as your reference angle to find 𝑥?
8
A
35°
55°
x
C
(c) Complete the statement to make a true statement:
sin 35° = cos _________. Explain your reasoning.
4. Timmy parents paid to repave the slanted driveway so that Timmy could ride
his bike up safely. Following the city restrictions, they made sure that the
driveway made a 75° angle with the ground. If Timmy’s house is 3 feet higher
than the road, what is the approximate length of the driveway?
17
5. Will the side lengths 89, 15, and
4√481 form a right triangle?
Explain your reasoning.
5.7
Word to Your Word Problems
How to handle word problems
The R word: READ each problem carefully circling/highlighting significant given values.
The S word: SKETCH the diagram if not given. Keep in mind we’re working with right triangles.
The L word: LABEL each value in the diagram, labeling the missing portion with a variable.
1. The batter hits the ball at an angle of 52° into the air. If
the ball reaches a height 7.6 feet above the pitcher, then
what is the distance from the pitcher to the batter?
2. A 13 foot ladder leans against a wall. If the angle the
base of the ladder makes with the ground in 25°, then
find the height the ladder reaches up the wall.
52°
● Angle of ELEVATION: When you
look up at an object, the angle that your
line of sight makes with a line drawn
horizontally.
● Angle of DEPRESSION: When you
look down at an object, the angle that
your line of sight makes with a line
drawn horizontally.
Example 1: Read, Label, and solve.
1. The angle of elevation from the base to the top of a slide is
about 13°. The height of the slide is 13.4m. Estimate the
length of the slide.
2. Emily is snowboarding down the black diamond run. If
the angle of depression is 52°, then how far does she
snowboard down the mountain?
1200m
3. A sonar operator on a cruiser detects a submarine at a
distance of 500 m and an angle of depression of 37°. How
deep is the submarine?
18
4. Larry the Ladybug is standing 14 foot from a tall tree.
The angle of depression from the tree to Larry is 43°.
Find the height of the tree.
5. Alex is going sky diving. The plane takes off at an angle of 26° with the ground. If Alex free falls vertically to the
ground and lands 285 feet from where the plane took off. At what altitude did Alex jump from the plane?
300 yds
6. Joshua is at the corner of Penn Street and Greenleaf Avenue. He needs to get to the corner of Bright Avenue and Mar
Vista Street. How much longer will he have to walk if he takes Greenleaf avenue and Mar Vista street than if he could
walk directly from one corner to the other (without taking streets)?
61°
A Step Up….
7. Jane bounce passed the basketball to Emily.
Determine the distance from Jane to Emily.
Emily
Jane
5.5 ft
3 ft
70°
75°
1
1 ft
8. You are a block away from a skyscraper that is 780 feet tall.
Your friend is between the skyscraper and yourself. The
angle of elevation from your position to the top of the
skyscraper is 42°. The angle of elevation from your friend’s
position to the top of the skyscraper is 71°. To the nearest
foot, how far are you from your friend?
ft
2
9. The baseball player accidentally nailed a bird with a
grounder. What is the total distance the ball travels from
the baseball player’s bat
•
•
5 ft •
• •
12 ft
10. (a) Are the triangles below are similar? Explain your reasoning.
19
•
•
• 51°
•
•
10 ft
A
50°
(b) Find the perimeter of quadrilateral ADCB.
P
11. Find the perimeter of each quadrilateral below.
a. Perimeter of 𝑍𝐴𝐶𝐾
Z
23
b. Perimeter of 𝐽𝐾𝐿𝑀
J
34
A
18
25
K
24
60°
C
K
M
23
45°
N
L
A
18
60°
C
Review: Partner Up!
1. If the hypotenuse of a 30°-60°-90° triangle is 8 feet, then find the length of
the longer leg.
2
3. Simplify (3√5)
2. Simplify
4
.
√2
4. Renee is debating whether or not she should drive to the gym and run 5 miles or just run
from her house to the gym and back. Which workout would cover more miles? Explain your
reasoning.
Gym
24 miles
21
42
63°
x x59°
13
70°
Renee's House
25 miles
Work
20
5.8
Similar Right Triangles + Perimeter Problems
3
Opening Question: Are all right triangle similar to each other? Explain.
3
45°
I. Similar or Not?
Example#1: (a) Determine whether each pair of right triangles are similar or not. Show work.
(b) Which of these sets of triangles are special right triangles?
J
a. A
b.
c.
3
I
D
3
F
B
E
L
J
3
I
F
P
O
2
4
4
3
Q
45°
C
G
45°
K
H
M
4√3
K
N
H
G
L
O
2. (a) Why are the 4triangles
4 similar?
4
P
2
2
4
Q
(b) Find the length of ̅̅̅̅
𝐵𝐷.
M
P
N
A
Q
10
N
D
C
B
5
13
E
(c) Classify each triangle.
(d) Should the classifications be the same?
Why or why not.
10
B
3. Given ̅̅̅
𝐽𝐹 bisects ∠𝐺𝐹𝐼 and 𝑚∠𝐺𝐹𝐼 = 68°.
F
(a) Why are the triangles similar?
8
(b) Find the length of ̅̅̅
𝐹𝐽.
10
G
H
J
21
I
6. Circle the similar triangles below. Show your work.
a. A
b.
F
I
c. I
D
F
D
G
4
C
5
C
L
L30°
L
J
L
K
50°
E
J
d.
J
J
E
5√3
B
G
65°
F
I
2
G
G
K
K
K
5
25°
H
H
7. Which information would show the two right triangles are similar? Choose allH
that apply.
E
A.
B.
C.
D.
sin 𝐺 = cos 𝐻
sin 𝐸 = sin 𝐻
tan 𝐽 = sin 𝐸
̅̅̅̅
̅̅̅̅
𝐸𝐺 ≅ 𝐻𝐽
H
H
G
F
J
I
8. Find all the sides of the two right triangles. Are they similar or congruent?
A
48°
8
8
D
48°
12
12
32°
32°
10 F
E
10
C
B
C
B
9. Find all the sides of the two right triangles. Are they similar or congruent?
20
J
K
53.13°
G
7.5
H
I
10
L
10. Given three triangles which statements are true?
Choose all that apply.
S
9
J
A. ∆𝑆𝐼𝑀~∆𝑇𝑂𝐽~∆𝑈𝑉𝐿.
I
U
B. tan 𝑀 = tan 𝑇.
9√3
C. 𝐽𝑂 = 18√3
60°
30°
18
L
D. cos 𝑆 = sin 𝐿
E. ∆𝑇𝑂𝐽 ≅ ∆𝑆𝐼𝑀
𝐽𝑂
F. 𝑂𝑇 =
M
O
√3
1
22
T
V
6