Download Applications of the Converse USES OF THE PYTHAGOREAN

4.1
Name __________________________________ Per______
Converse of the Pyth TH and Special Right Triangles
CONVERSE OF THE PYTHAGOREAN THEOREM
, then ABC is a
If
Can be used to check if a figure is
a right triangle.
_________ triangle.
Example 1: Tell whether the given side lengths would form a right triangle.
b) 6, √
a) 29, 24, 16
,9
Applications of the Converse
(2 LEGS)
LONGER LEGS
SHORTER LEGS
(HYP)
• If a2 + b2 = c2, then ABC is a _________ triangle.
• If a2 + b2 < c2, then ABC is ________. (shorter legs)
• If a2 + b2 > c2, then ABC is _________. (longer legs)
A cute super model
obese (obtuse)
Example 2: Classify the triangles as right, acute, or obtuse.
b) 5, 4, √
a) 17, 4, 14
c) 16, √ , 8
USES OF THE PYTHAGOREAN THEOREM
When you already know it’s a RIGHT triangle…
Given 3 sides lengths of A triangle…
Given 2 side lengths → find the 3rd side length
Plug into Pyth. Thrm → find out if it’s right, acute,
or obtuse
Neat things you can do with proportions (will come in handy later )
The “Flip”
If
The “Swap”
, then
If
1
, then
45°-45°-90° TRIANGLE THEOREM
45° - 45° - 90°
In a 45-45-90 triangle, the extended ratio
45°
c
b
of the sides is 1: 1:
:
45°
* Note: Legs are always __________________
:
√
a
Ex#1: Find the value of the variables. Leave answers as simplified radicals.
Is this a
45-45-90 ∆?
Why?
2. If z = √ , then find the value of x. 3. What is the value
of x in the triangle?
1. Solve for .
45°
x
10
30°-60°-90° TRIANGLE THEOREM
In a 30-60-90 triangle, the extended
ratio of the sides is 1: √ : 2
a
30° - 60° - 90°
6 °
c
°
:
√3
b
Ex#2: Find the indicated length. Leave answers as simplified radicals.
1. Solve for .
3. Solve for and .
2. If c = √ , find the values of a
and b.
𝑐
15
2
:
Example 3: In groups of two find the missing sides indicated in each problem. If possible,
leave answers in simplified radicals.
2. Which would serve as a counterexample to the
1. If
, then find the value of .
statement below?
x
“Any three side lengths of a triangle forms a right
y
triangle.”
30º
z
A. 2, 9, 5
A. 11
B. 3, 4, 5
B. 11√
C. 21, 20, 19
C. 22√
D. 11, 61, 60
D. 11√
3) Find the value of n and m.
4) Find the value of k.
20
18
n
k
55°
14
m
*Problems that are easily confused…What to Use? What to Use?
1. In the figure below, n is a whole
number. Find the value of n.
Explain.
n
2. In the figure below, n is a whole
number. What is the smallest
possible value of n? Explain.
n
n
n
20
5
n
20
5
k
k
42
3. In the figure below, n is a whole
number. What is the largest
possible value of n? Explain.
55°
55°
n
14
14
14
A. 4
A. 6
A. 9
B. 8
B. 7
B. 10
C. 16
C. 8
C. 11
D. 2
D. 9
D. 12
3
4.2
More Special Right Triangles
RECTANGLE
SQUARE
EQUILATERAL TRIANGLE
A diagonal divides a rectangle
into two _________________ ∆s.
Use Pythagorean Theorem
A diagonal divides square into
two ____________ ∆s.
The altitude (height) divides it
into two ________________ ∆s.
Use ratios _______________
 Use ratios _______________
Ex#3: If necessary in the problems below, leave your answer as a simplified radical.
2. ΔHAT is an equilateral triangle, find the length of its
1. YELP is a square and its diagonal has a length 18√
altitude.
inches. Find the length of one of its sides.
Y
A
E
10
P
L
H
T
b. Find the area of YELP.
b. Find the area of ΔHAT.
3. If the length of the diagonal of a rectangle is 39 inches
and the width of the rectangle is 15 inches, then find
the area of the rectangle. (Draw a st inkin’ picture!)
4. Find length of the diagonal of a square with whose
perimeter is 32 feet. (Draw a stinkin’ picture!)
5. Find the side length of an equilateral triangle whose
height 21meters. (Draw a stinkin’ picture!)
6. You are standing 18 feet from a building. The vertical
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°
4
REVIEW
1. Classify a triangle with side lengths 14, 19, and
√ .
A. right
B. acute
C. obtuse
D. not a triangle
2. If j = 6 in the right triangle below, then find the value
of k.
A.
B.
C.
D.
k
6
6√
12
12√
j
60°
60°
3. Find the perimeter of an equilateral triangle whose
altitude (height) is √ feet.
4. Solve for x and explain.
A.
B.
C.
D.
A. 9
B. 18
C. 54
D. 6√
l 60°
73
53
56
50
28
x
45
SPIRAL REVIEW - Transformations
THE RULES…
Preimage (1, 3)
REFLECTION in the x-axis
Preimage (1, 3)
ROTATION 90˚,
ROTATION
REFLECTION in the y-axis
˚
(1,3)
(1,3)
1. The point A(4, -2) is transformed to A’(2, 4). Name
the transformation.
2. The point B(4, -2) is transformed to B’(4, 2). Name
the transformation.
a.
b.
c.
d.
3. The point H(1, 5) is transformed to H’(-5, 1). Name
the transformation.
a.
b.
c.
d.
a.
b.
c.
d.
Reflection across the x-axis
Reflection across the y-axis
90˚ rotation clockwise about the origin
90˚ rotation counterclockwise about the origin
Reflection across the x-axis
Reflection across the y-axis
90˚ rotation clockwise about the origin
180˚ rotation about the origin
4. The point G(8, -9) is transformed to G’(-8, 9). Name
the transformation.
Reflection across the x-axis
Reflection across the y-axis
90˚ rotation clockwise about the origin
90˚ rotation counterclockwise about the origin
a.
b.
c.
d.
5
Reflection across the x-axis
Reflection across the y-axis
90˚ rotation clockwise about the origin
180˚ rotation about the origin
SPIRAL REVIEW - Area and Perimeter
Perimeter
REFRESH
your
memory!
Area of a
Rectangle
1. Find the perimeter of the figure below.
Area of a
Triangle
Area of a Circle
Circumference
3. Find the area of the triangle below.
(All line
segments meet at right angles).
12
3
10
5
8
23
7
9
Lesson Preview
1) Label the hypotenuse H
2) Put a smiley face on the side adjacent to
3) Put a star on the side opposite of
A
Question: Would your answer change if you labeled the side opposite of
Why or why not?
and adjacent to
Practice: Label an O for opposite side, A for adjacent side, and an H for hypotenuse based on the given angle.
1. Label from
2. Label from
3. Label from
I
R
U
M
R
G
K
G
I
*Which side is always labeled the same regardless of what angle you start from? O, A, or H? Why?
6
?
4.3 Intro to Trigonometric Ratio
Identifying the Hypotenuse, opposite, and adjacent side of a right△
Reference angle
Hypotenuse
Opposite
1. Using J as your reference angle, label the
hypotenuse, opposite, and adjacent sides.
Adjacent
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
How to set it up! The sine, cosine, and tangent of DEF from reference angle D
E
F
D
*Each trigonometric ratio depends on which acute angle you are starting from…
Here’s How to MEMORIZE the TRIG RATIOS
SOH
CAH
TOA
cos ∠A =
sin ∠A =
tan ∠A =
TYPE 1: Given side lengths of a right triangle, find the sine, cosine, and tangent
1.
S
a.
b.
c.
d.
58
42
T
40
2. Find the trigonometric ratios of each right
triangle.
B
3. Find the trigonometric ratios of each right
triangle.
J
5
C
12
the same as
R
16
A
√
K
7
8
L
4. Find the sinA if cosA =
5. Find the cosG and sinG if tanG =
P
.
A
I
C
B
G
TYPE 2: Given 1 SIDE and 1 ACUTE ANGLE, find a side length
Steps to solve:
1) Pick a given ACUTE angle to start from (Mark it up and label OPP, ADJ, HYP)
2) Use SOH-CAH-TOA (pick 2 sides for ratio -1 you WANT & -1 you HAVE)
3) Cross multiply and solve! *Don’t round until the end!
Find the missing side indicated. Round any values to the tenths place.
1.
2.
8

3
.
4.4 Applying Trigonometric Ratios
USING TRIGONOMETRIC RATIOS
1. Mark your reference angle
14
x
2. Label the Hypotenuse, the Opposite and Adjacent side
(*based on given acute angle)
35°
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
4. Set up your trig ratio (SOH-CAH-TOA) and solve
Example 1: Use Trigonometric Ratios to Find Side Lengths
Find the value of x. Round to the nearest tenth.
1.
2.
34
x
x
3. Find the lengths of both missing sides. Round to the nearest tenth.
72º
12
TO FIND THE MISSING SIDE OF A RIGHT TRIANGLE
GIVEN 2 SIDES…
GIVEN 1 SIDE AND 1 ANGLE…
• Use the Pythagorean Theorem
• If it’s as SPECIAL right ∆
Ex:
15
11
45°-45°-90°  1: 1: √
If it’s NOT a special one…
• Use a Trigonometric Ratio
Ex:
63º
30°-60°-90° 1: √ : 2
11
SOH – CAH – TOA!
9
Example 3: Find the value of x. If possible leave your answers as a simplified radical or
round to the nearest tenth.
1.
42
x
2.
15
A.
70º
B.
x
9
4
C.
D.
3. Find x.
4
4. If b = 14 in the right triangle below, then what is the
value of c?
x
c
b
60°
6
22°
a
A. 6√
B. √
C. 12
D. 3
g
5. Find the value of f and g.
6. Timmy rides up his 12-foot high slanted driveway.
Estimate the length of the driveway.
g
82
f
56°
28
34°
j
82
h
More review
1.
G
15
10
I
5 13
a) Find tan H
2. Find the sin B and cos B if tan B
A
C
b) Find cos H
H
10
B
4.5 More Word Problems + Multiple Choice
Word Problems…Can you say GLORIOUS? 
● 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: Solve the word problems. Round to the nearest tenth.
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. To an observer on a cliff 360 m above sea level, the
angle of depression of a ship is 28°. What is the
horizontal distance between the ship and the base of
the cliff?
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?
4. A 14-foot ladder makes a 76° angle of elevation with
the ground. How far up the wall is the top of the
ladder?
sin 76°
cos 76° .24
tan 76°
5. Larry the Ladybug is standing 14 ft from a tall tree. The angle of elevation from the ground to Larry is 43°. Find the
height of the tree.
11
A Step Up….
6. Jake’s eye level is feet above the ground From Jake’s eye level, the angle of elevation is to the top of a building is
29° If the distance from Jake’s eye level to the top of the building is 5 feet, how tall is the building?
7. Sally went to the zoo and stood 1 feet away from a giraffe The angle of elevation from Sally’s eye level to the top
of the giraffe’s head is °. If the giraffe is 12 feet tall, then find the distance
from Sally’s eye level to the ground
8. A crazy person was chasing Zack down the street The angle of elevation from Zack’s eye level to the top of the
crazy person’s head is °. If the crazy person is 6 foot tall and the distance from Zack’s eye level to the ground is 4,
then how far away is Zack from the crazy person?
In groups of 2.
1. Find the perimeter of the triangle below. Round your
answer to the tenths place.
2. Find the area of the triangle below. Round your answer
to the tenths place.
14
30°
14
64°
6
12
How To Handle All Types of Right Triangle Problems…
Finding SIDE LENGTHS in a Right Triangle
Is it special???
___ - ___ - ___ or ___ - ___ - ___
YES!
45˚ - 45˚ - 90˚
:
NO!
Given:
2 sides lengths
30˚ - 60˚ - 90˚
:
:
Given:
1 angle and 1 side length
:
___________________
____________
45°
30°
______– ______– ______
H
O
45°
60°
c
a
H
O
A
A
b
30°
RIGHT TRIANGLES in POLYGONS
H
• Rectangle w/ a diagonal
O
→ ______________________
A
30°
H
• Square w/O a diagonal
60°
→ ______________________
A
• Equilateral ∆ w/ altitude
60°
→ ______________________
60°
60°
60°
Classifying a Triangle as RIGHT, ACUTE, or OBTUSE…
Right Triangle
Obtuse Triangle
Acute Triangle
HINTS!
* Make sure “c” is the largest
number
legs = hyp
a2 + b2 = c2
legs < hyp
a2 + b2 < c2
legs > hyp
a2 + b2 > c2
13
* If √
, then use decimals
to check size only
O
H
A
Examples
1. Find the value of x and y.
°
𝑥
17
O
3. If
, then find
14 3
5 °
15
A
15
𝐾
x
60°
4. Find sin V .
U
2. Solve for x.
H
5. Find the height of an equilateral
triangle with a side length of 12
6. Solve for x.
x
28○
V
16
17
W
6
A.
B.
H
14 3
60°
7. If the diagonal of a square is 10, then find the area of
the square.
A.
B.
C.
D.
6
C.
x
D.
8. Find the measure of
and explain.
R
100
25
50
200
U
A.
B.
C.
D.
14
15˚
17˚
120˚
35˚
G
.
4.6 Inverse Trigonometric Ratios
Opening Question: Are the questions below asking for the30°same thing?
1. Find sin A .
2. Find mA .
A
12
32
60°
60°
B
C
60°
What’s an Inverse Operation?
H
14and
3 subtraction
Inverse Operations are two operations that undo each other.O For example, addition
are inverse operations you use in order to get a variable by itself. A
x
60°
28 Ex: x x + 3 = 5
45
To get rid of trig ratios to find an angle use the inverse operations of sine, cosine, and tangent …
♦
♦
♦
So in general: To solve for
…
( )
(
)
( )
(
( )
)
°
( )
(
( )
)
( )
°
°
Example 1: Find the measure of each given angle. Round to the nearest tenths place.
sin A  0.85
1.
tan R  1.05
2.
3.
Example 2: Find the measure of B. Round your answer to the nearest tenth.
1. C
2. A
A
3.
30
8
14
A
C
19
22
B
C
B
Find the missing sides and angles. Round answers to the nearest tenth.
4.
15
40
B
Trig. Ratios VS. Inverse Trig. Ratios
• Use trigonometric ratios when trying to find a
missing side given an angle and a side.
• Use INVERSE trigonometric ratios when
trying to find a missing angle given two sides.
Ex:
Ex:
tan 37º =
?
(tan -1)tan A = (tan -1)
6
6
37º
tan A =
?
3
A
SOLVE A RIGHT TRIANGLE
A = tan -1
A
8
?
To solve a right triangle, find the measures of all _________
?
?
?
and all __________.
?
When given 1 side & 1 angle
(besides the right
angle)
When given 2 sides
1. Pick a reference angle, and use a TRIG RATIO
to solve for a missing side.
• (sin, cos, or tan)
2. Use a TRIG RATIO or PYTHAGOREAN TH
to find the last side.
• (sin, cos, or tan) OR (a2+b2 =c2)
3. Find the last angle
Use TRIANGLE SUM TH to find the measure
of the missing angle.
•(
°)
(besides the right angle)
1. Use an INVERSE TRIG RATIO to find the
measure of a missing angle.
• (sin-1, cos-1, or tan-1)
2. Use TRIANGLE SUM TH to find the measure
of the last angle.
•(
°
3. Find the last side
Use PYTHAGOREAN TH to find the missing
side
• (a2+b2 =c2)
**Hint!** Start with what you have 2 of. TRIG ratio for what you have 1 of.
Example 1: Solve the right triangle. If possible, leave your answer as a simplified radical or round to the
nearest tenth if necessary.
1.
2.
16
3.
2.
21
EXTRA PRACTICE If possible, leave your answer as a simplified radical or round to the nearest tenth.
1. Solve a right triangle that has a 30º angle and a 14 inch hypotenuse. (DRAW IT!!!)
2. Solve a right triangle with legs that measure 16 and 24 ft. (DRAW IT!!!)
17
6
4.7 Spiral Review
Parallel lines
Corresponding s
REFRESH
your
memory!
Consecutive Interior s
Alternate Interior s
Alternate Exterior s
When lines are
...
1. Solve for x. Explain.
2. Solve for x. Explain.
x˚
46˚
58˚
x˚
50˚
3. Solve for x. Explain.
4. Solve for x. Explain.
78˚
98˚
y˚
x˚
61˚
32˚
x˚
Similarity
SSS ~
REFRESH
your
memory!
SAS ~
S
L
S
AA ~
L
L
L
M
S
S
M
1. Use Angle-Angle Similarity Postulate to determine which pair of triangles is not similar.
a.
b.
c.
d.
˚
˚
˚
˚
˚
b
˚
c
a 60°
28
M
N
2. Use Side-Angle-Side Similarity to determine which pair must be similar.
a.
b.
c.
4
8
6
4
U
3
10 D
63°
12
6
5
A
15
8
M
c
a 60°
28
M
5
7
N
18
4
14
94°
11.2
3
94°
20
11.2
7
6
b
d.
8
8
1st: Write Similarity Statement
Finding Missing
Sides of Similar
Triangles
11.2
2nd: Find Ratio of Similarity
7
3rd: Set up proportion & Solve
1. If AN = 9, HO= 26, and AT = 16, find YO. Do not leave
your answer in a decimal.
11.2
2. IF OU = 56, UG = 14, YU = 4, and YG = 18, find DO.
Do not leave your answer in a decimal.
N
O
O
T
D
U
A
Y
H
Y
G
3. If JM = 12, MS = 15, and AM = 18, find ME. Do not
leave your answer in a decimal.
4. If VR = 18, VI = 6, and IT = 8, find IC. Do not leave
your answer in a decimal.
J
I
V
C
E
M
T
A
J
R
S
E
M
11.2
7
A
S
O
Triangle Sum
All the interior angles of a triangle add up
U to _______
1. What is the measure of the largest angle of the
triangle?
Y
2. Find the measure of the smallest angle.
G
(x + 36)˚
(4x + 9)˚
(3x – 17)˚
19
Classifying
triangles
Acute ∆
Scalene ∆
Isosceles ∆
Equilateral ∆
Obtuse ∆
Right ∆
Equiangular ∆
Classify the triangle by its angles and sides.
1.
2.
3.
70°
34°
60°
55°
52°
60°
If (hypothesis), then (conclusion)
Converse
Counterexample
True for 1st part
False for 2nd part
1. “If a triangle has two congruent sides, then it is a right triangle.”
a. Write the converse of the statement
b. Determine whether each example below serves as a counterexample. Explain why for each.
Ex#1: 14, 14, 10
Ex#2: 6, 6, 6√
2. “If two angles are acute, then they are congruent.”
a. Write the converse of the statement
b. Determine whether each example below serves as a counterexample. Explain why for each.
Ex#1:
Ex#2:
°
°
20