Download 7.1/7.2 Apply the Pythagorean Theorem and its Converse Find

7.1/7.2 Apply the Pythagorean Theorem and its Converse
Remember what we know about a right triangle:
In a right triangle, the square of the length of the ________________ is equal to the sum of the
squares of the lengths of the _____________.
_______ +_______ = ________
Remember that a and b
represent the lengths of the
legs and c always represents
the length of the hypotenuse.
There are many different methods to prove the Pythagorean Theorem. One method is to use the
distance formula.
(4, 7)
c
(1, 3)
(4, 3)
Find missing sides using the Pythagorean Theorem:
1.
2. Find the area of the following triangle.
3. A 5 foot board rests under a doorknob and the base of the board is 3.5 feet away from the bottom
of the door. Approximately how high above the ground is the doorknob?
Pythagorean Triple: ___________________________________________________
___________________________________________________________________
Pythagorean Converse:
Use the converse to determine if a triangle is ______________, _____________ or _____________.
Triangle Inequality Theorem: ________________________________________________________
________________________________________________________________________________
Classify the triangle as acute, obtuse, or right
4.
5. 22, 26, 14
6.
7.3 Similar Right Triangles
Identify the similar triangles in the diagram. Sketch the three similar right triangles so that the
corresponding angles and sides have the same orientation.
Corresponding Sides of Similar Triangles are in Proportion
EXAMPLES:
17: ___________________________
20: ___________________________
y: ____________________________
Find the values of the missing variables after you decide which proportion to set up (leg or alt).
1.
2.
3.
7.4 Special Right Triangles
Radical Review: Simplify the following radicals. NO DECIMALS!
1.
2.
3.
A 45 -45 -90 triangle is an isosceles right triangle that can be formed by cutting a square in half.
Find the missing sides of each 45 -45 -90 triangle.
4.
5.
6.
7.
8.
9.
45̊
15
45̊
A 30 -60 -90 triangle is a right triangle that can be formed by cutting an equilateral triangle in half.
Find the missing sides of each 30 -60 -90 triangle.
10.
11.
12.
13.
14.
15.
7.5/7.6 APPLY THE TANGENT, SINE, AND COSINE RATIOS
TRIGONOMETRY: USED TO FIND MEASURES IN TRIANGLES
VOCABULARY
TRIGONOMETRIC
RATIO
TANGENT RATIO
SINE RATIO
COSINE RATIO
SOH CAH TOA
When we define our trigonometric ratios, we label each side of a right triangle as opposite leg (opp), adjacent
leg (adj.), or hypotenuse with respect to which angle we are using. Sometimes we use the symbol theta, , in
place of x for an angle.
Do you remember
Example: Label each side as opposite, adjacent, or hypotenuse with respect to .
how to find a missing
side of a right
triangle?
_______________
Find the three trig ratios for each angle.
a 2  b2  c 2
sin C = __________
sin B =___________
cos C = __________
cos B =___________
tan C = __________
tan B = ___________
A
C
sin A = __________
sin B = ___________
cos A = __________
cos B = ___________
tan A = __________
tan B = ___________
B
Use the trig functions to help find missing sides of a triangle.
You can use your calculator to evaluate trig functions of any angle. Use the keys,
Before using the calculator, make sure it is set in DEGREE MODE.
.
Use the inverse trig functions to help find missing angles of a triangle.
When trying to find a missing angle, use your trig inverse button by pressing the 2ND button before
. ( sin 1 , cos 1 , and tan 1 )
Use your calculator to estimate the measure of A to the nearest tenth of a degree.
sin A = 0.76
cos A = 0.17
tan A = 0.9
Use a calculator to approximate the measure of A to the nearest tenth of a degree.
Solve the right triangle.
(Find all missing parts)
Angle of Elevation and Angle of Depression:
VOCABULARY
If you look up at an object, the angle your line of sight makes with a horizontal line is called
the angle of elevation.
EXAMPLE:
ANGLE OF
ELEVATION
Railroad A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of
35° Find the lengths x and y
If you look down at an object, the angle your line of sight makes with the horizontal line is
called the angle of depression.
EXAMPLE:
ANGLE OF
DEPRESSION
Roller Coaster You are at the top of a roller coaster 100 feet above the ground. The angle
of depression is 44°. About how far do you ride down the hill?
Notice how the angle
of angle of elevation is
always equal to the
angle of depression.
Do you remember why
from first semester?
Solve the following story problems. Draw your own picture for each.
You attend a music concert with some friends and sit
halfway up the bleachers in the arena. The angle of
depression from your horizontal line of sight to the
stage is 24°. If your seat is 45 feet above stage level,
what is your actual distance d from the stage?
You are standing 350 feet away from a skyscraper
that is 750 feet tall. What is the angle of elevation
from you to the top of the building?
Chapter 7 Extension
Law of Sines and Law of Cosines
Review:
Right Triangles
Non-right Triangles
Pythagorean Theorem
SUM OF ANGLES OF A TRIANGLE =180
Special Right Triangles
(45-- 45-- 90) and (30-- 60-- 90)
SOH CAH TOA
*LAW OF SINES
*LAW OF COSINES
Notice that
A and a
B and b
C and c
are all
opposite each
other.
Examples:
1)
x = ________
2)
x = ________
3) Solve the triangle.
AC = ________
b
AB = ________
B = ________
4) Solve the triangle.
AC = ________
C = ________
B = ________
b
Examples:
1)
x = ________
2)
x = ________
3) Solve the triangle.
AC = ________
A = ________
C = ________
4) Solve the triangle.
B = ________
C = ________
A = ________