Download Students Will Be Able To

Name _________________________________________________________ Hour __________
Chapter 9
Geometry
1
Warm-up:
Students Will Be Able To use the Pythagorean Theorem to find missing information on
right triangles.
Pythagorean
Theorem
9.2 Pythagorean Theorem
Theorem 9.4
a2 + b2 = c2
Where a and b are legs of a right triangle, and c is
the hypotenuse.
c
b
a
Pythagorean
Triple
Special case of a right triangle where a, b and c are
all positive integers
Area of triangle A = ½ b h (for any triangle)
Examples:
Find the unknown side length. Simplify answers
that are radicals. Tell whether the side lengths
form a Pythagorean triple.
12
7
1.
2.
x
5
x
4
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
2
Examples:
Find the height of the triangle. Then find the area
of the triangle. Round decimal answers to the
nearest tenth.
7m
h
7m
10 m
Homework: 9.2 wkst
Warm-up:
Students Will Be Able To determine the type of triangle using the Pythagorean Theorem
Triangle
Inequality
Theorem
Pythagorean
Theorem
Converse
9.3 The converse of Pythagorean Theorem
The sum of the lengths of two sides of a triangle is
greater than the length of the third side.
C
A
B
If c2 = a2 + b2, then ABC is a ___________________
If c2 < a2 + b2, then ABC is an __________________
If c2 > a2 + b2, then ABC is an __________________
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
3
Examples:
The triangles below appear to be right triangles.
Tell whether they are right triangles.
1.
8
7
113
2.
4 95
15
36
Decide whether the numbers can represent the side
lengths of a triangle. If they can, classify the
triangle as right, acute, or obtuse.
3. 38, 77, 86
Homework: 9.3 wkst.
Warm-up:
Students Will Be Able To demonstrate a knowledge of special right triangles.
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
4
9.4 Special Right Triangles
450-450-900
Triangles
Hypotenuse = leg times √2
45
leg 2
legx
45
x
leg
300-600-900
Triangles
Hypotenuse = 2 times short leg
Long leg = short leg times√3
60
2xShort leg
x
Short leg
30
long leg 3
Examples:
Find the value of x.
1.
3
3
45o
2.
5
x
x
x
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
5
Examples:
Find the value of a and b.
3.
60o
b
a
30o
5
Homework: 9.4 wkst.
Warm-up:
Students Will Be Able To demonstrate a knowledge of special right triangles.
Examples:
9.4 More Special Right Triangles
Find the value of a and b.
1.
60o
12
b
30o
a
Homework: 9.4 worksheet
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
6
Warm-up:
Students Will Be Able To identify and use similar triangles among right triangles.
Theorem 9.1
9.1 Similar Right Triangles
If the altitude is drawn to the hypotenuse of a right
triangle then the two triangles formed are similar to
the original triangle and to each other.
C
A
D
B
Theorem 9.2
In a right triangle, the altitude from the right angle
to the hypotenuse divides the hypotenuse into two
segments.
The length of the altitude is the geometric mean of
the lengths of the two segments.
𝐶𝐷 = √𝐴𝐷 ∙ 𝐷𝐵
Theorem 9.3
In a right triangle, the altitude from the right angle
to the hypotenuse divides the hypotenuse into two
segments.
The length of each leg of the right triangle is the
geometric mean of the lengths of the hypotenuse
and the segment of the hypotenuse that is adjacent
to the leg.
𝐶𝐵 = √𝐴𝐵 ∙ 𝐷𝐵
𝐴𝐶 = √𝐴𝐵 ∙ 𝐴𝐷
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
7
Examples:
a.
Identify the similar triangles
Y
5.5 m
3.1 m
h
Z
W
6.3 m
b.
Find the height of the right triangle.
Find the value of each variable.
a.
x
6
b.
2
y
Homework: 9.1 wkst
3
5
X
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
8
Warm-up:
Students Will Be Able To identify and use similar triangles among right triangles.
Examples:
9.1 Similar Right Triangles
Find the value of each variable.
a.
14
x
10
4
b.
z
7
Homework: 9.1 wkst
Warm-up:
Quiz 9.1-9.4
8
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
9
Warm-up:
Students Will Be Able To use trigonometric ratios to solve for missing sides of right
triangles.
Ratios
Examples:
9.5 Trigonometric Ratios
Let ABC be a right triangle. The sine, cosine, and tangent
of the acute angle are defined as follows:
B
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ∠𝐴 𝑎
𝑆𝑖𝑛 𝐴 =
=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐
c
a
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠𝐴 𝑏
𝐶𝑜𝑠 𝐴 =
=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐
b
C
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ∠𝐴 𝑎 A
𝑇𝑎𝑛 𝐴 =
=
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠𝐴 𝑏
Find sine, cosine, and tangent of the following triangle.
Sin A =
B
Cos A =
50
Tan A =
A
Sin B =
Cos B =
Tan B =
48
14
C
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
10
Use a calculator to approximate the given value to four
decimal places.
1.
Sin 48o
2. Cos 1240
Homework:
9.5 wkst
Warm-up:
Students Will Be Able To use trigonometric ratios to solve for missing sides of right
triangles.
9.5 More Trigonometric Ratios
450-450-900 Triangles
Special
Triangles
45
Sin 450 =
2
1
Cos 450 =
45
Tan 450 =
1
300-600-900 Triangles
60
2
1
3
Sin 300 =
30
Cos 300 =
Tan 300 =
Sin 600 =
Cos 600 =
Tan 600 =
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
11
Angle of
Elevation
The angle from the ground to the line drawn from a
point on the ground to the top of the object
(hypotenuse)
Example:
Find the height of the tree.
h
590
45 ft
Homework: 9.5 wkst.
Warm-up:
Students Will Be Able To use trigonometric ratios to solve for missing sides of right
triangles.
9.6 Using trig functions to solve triangles
Inverse Trig
Identities
Sin A =
Cos A =
Tan A =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
opposite
 mA
hypotenuse
adjacent
Cos 1
 mA
hypotenuse
opposite
Tan 1
 mA
adjacent
Sin 1
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
12
Examples:
Solve each triangle for the missing information.
1.
10
8
b
Solve each triangle for the missing information.
2.
S
15
r
20o
R
Homework: 9.6 wkst.
Warm-up:
s
T
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
13
Students Will Be Able To use trigonometric ratios to solve for missing sides of right
triangles.
Examples:
9.6 More Triangles
∠A is an acute angle. Use a calculator to
approximate the measure of ∠A to the nearest
tenth of a degree.
1. tan A = 0.5
2.
sin A = 0.35
Solve each triangle for the missing information.
S
1.
15
r
32o
R
Homework: 9.6 worksheet
s
T
Name _________________________________________________________ Hour __________
Chapter 9
Geometry
14
Warm-up:
Quiz 9.5-9.6
Warm-up:
Warm-up:
Warm-up:
Chapter 9 Test