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Geometry
of Right Triangles
Pythagorean Theorem
Angles of Elevation and Depression
Law of Sines and Law of Cosines
Recall that a right triangle is a triangle with a right angle.
The sides form that right angle are the legs.The side opposite the right angle is the hypotenuse.
The hypotenuse is also the longest side.
hypotenuse
In a right triangle, the sum of the squares of the lengths
of the legs is equal to the square of the length of thehypotenuse.
Pythagorean Theorem
Example:
Find the length of the missing side of the right triangle.
Is the missing side a leg or the hypotenuse of the right triangle?
81 + 144 = x
-15 is a extraneous solution, a distance
Solve for x:
Is the missing side a leg or the hypotenuseof the right triangle?
Example:
Find the length of the missing sideof the right triangle.
Find the length of the missing side.
Find the length of the missing side.
The safe distance of the base of the ladder from a wall it leans
ladder should be 7 feet fromthe wall. How far up the wall
will a ladder reach?
Solve using a2 2 2
The dimensions of a high school basketball court are 84'
long and 50' wide. What is the length from one corner ofthe court to the opposite corner?
A NBA court is 50 feet wide and the length from
one corner of the court to the opposite corner is
94 feet
(Round the answer to the nearest whole number)
The PythagoreanTheorem can alsobe used in figuresthat contain right
Find the perimeter of the square.
note: Before findingthe perimeter of the
of each side.
Start here:222
Find the area of the triangle.
the height of the triangle.
By definition, the altitude (or height) of an isoscelestriangle is the perpendicular bisector of the base.
Find the perimeter of the rectangle.
120 square feet
46 square inches
46 square feet
Find the perimeter of the square. (Round to the
nearest tenth)
If the square of the longest side of a triangle is equal to the
right triangle.
Converse of the Pythagorean Theorem
Rememberlongest side
DIf the square of the longest side of a triangle is greater thanthe sum of the squares of the other two sides, then the
triangle is obtuse.
Theorem
If the square of the longest side of a triangle is less thansum of the squares of the other two sides, then the triangle
Theorem Classify the triangle as acute, right, or obtuse.
not a triangle not a triangle
not a triangle
acute triangle
right triangle
Review
, then triangle is right.
, then triangle is acute.
, then triangle is obtuse.
How many do you know?
Pythagorean Theorem. How?
BD
click
click
BD
The altitude of a right triangledivides the triangle intotwo smaller triangles that aresimilar to the original triangleand each other.
Given:
Prove:
Reasons
is a right angle
is a right angle
is a right angle
ABC ~ CBD
Given
Given
Def of Perp Lines.2 lines that form art angle
All rt angles are
Reflexive Prop of
AA~
Def of Perp Lines
All rt angles are
Reflexive Prop of
AA~60
3030
6060
30
60
30
3060
BD
d
ec
b
a
ca
b d
b ea
Reasons
d
ec
b
a
Given:
Prove:
ABC is a right triangle.is an altitude.
Using the multiplication
multiply the equation by bc.
simplify
Using the multiplication
multiply the equation by ac
simplify
Altitude of a rt triangletheorem.
Altitude of a rt triangletheorem.
Definition of similartriangles.
Definition of similartriangles.
(1)
(2)
Reasons
d
ec
b
a
Given:
Prove:
ABC is a right triangle.is an altitude.
Distributive Property
Simplify
Using the addition property
Given
Substitution
Find the length of the altitude KI?
12
x
x
x
x
x
15 Which ratio is the ratio of corresponding sides? 16
What is an arithmetic mean? The sum of n values dividedby the number of values (n).
For more information click on this link:Arithmetic Mean vs Geometric Mean
of two positive numbers a
given a rectangle with sides a and b, find the side of thesquare whose area equals that of the rectangle.
Find the geometric mean of 8 and 14.
(only the positive value)
17 Find the geometric mean of 7 and 56. Write the
answer is simplest radical form.
A
B
C
D
18 Find the geometric mean of 3 and 48.
divides the the hypotenuse into two segments. The
BD
19 Find x.
100
20 Find x.
that is adjacent to the leg
BC DB
BD
D
21 Is PR a geometric mean between QR and SR? 22
23 24 Find y.
5 16
25 Find y.
3
18
24
None of the above
26 Find x.
two special right triangles.
triangle is an isosceles right triangle, where thehypotenuse is √2 times the length of the leg.
hypotenuse = leg(
Can you prove this?
Find the length of the missing sides.
Write the answer in simplest radical form.
Find the length of the missing sides of the right triangle.Find the length of the missing sides.
27 Find the value of x.
5
28 Find the value of y.
5
29 30
right triangle,the hypotenuse is twice thelength of the shorter leg
and the longer leg is3 times the length of the
shorter leg.
hypotenuse = 2(shorter leg)longer leg = 3(shorter leg)
This can be proved using anequilateral triangle.
A C
c=2x
a=x D
b
Blet a = x, c = 2x and b= BD
Find the length of the missing sidesof the right triangle.
opposite the smallest angle and the longest
side is opposite the largest angle.
HF is the shortest side
GF is the longest side (hypotenuse)GH is the 2nd longest side
HF < GH < GF
Find the length of the missing sidesof the right triangle. Example
The altitude (or height)divides the triangle into two
The length of the shorter leg is 7 ft.
The length of the longer leg is 7
84.87 square ft
30
60
31 Find the value of x.
2)/2
32 Find the value of x. 33 Find the value of x.
34 35
feet and rises at angle of 30
hypotenuse = 2(2.5)hypotenuse = 5
36 A skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45 . What is the height of the ramp? Round to the
nearest hundredth.
37 What is the length of the base of the ramp? Round
to the nearest hundredth.
38 39
Ever since the construction of the
tilted south and is at risk of falling
below 83 degrees, it is feared thetower will collapse.
A
B C
D
F
E
WHY?
Engineers can measure the angle of slant using
Engineers very carefully measure theperpendicular distance from a tower window
Then they measure the distancefrom the tower to points C, E or G.
Triangle Height Base Ratio Height / Base
ABC AC=50m BC=5m 50/5=10
DBE DE=30m BE=3m 30/3=10
FBG FG=20m BG=2m 20/2=10
Let's calculate the ratio's of the height to the basefor each right triangle.
Notice that all of the ratios are the same.WHY?
The ratio of height/base is also called theslope ratio (rise/run) or tangent ratio.
Click for interactive website to investigate.
is the reference angle,the side opposite isthe side adjacent is b.and the hypotenuse is c.
When is the reference angle,the side opposite is b.the side adjacent (or next to) B is a.and the hypotenuse is c.
opp
hyp
opp hyp
40 What is the side opposite to J? 41 What is the hypotenuse of the triangle?
42 What is the side adjacent to J? 43 What is the side opposite K?
A JL
B LK
C KJ
44 What is the side adjacent to K?
A JL
B LK
C KJ
is the ratio of the two sides of a
sinesin
Trigonometric Ratios
adjacent side
The 3 Trigonometric Ratios
This spells....
SOHCAHTOA or
which is a pneumonic to help you remember the sides of a right triangle (you'll need to remember the spelling).
θ
ExampleD
6
8 4
6 3
D
opp
adj
hyp
ExampleD
8
6 3
8 4
D
opp
adjhyp
45 What is the sin R?
A 9/13
B 7/9
C 7/13
D
46 What is the cosR?
47 What is the tanR? 48 What is the sinQ?
A 9/13
B 7/9
C 7/13
D 9/7
49 What is the cosQ?
A 9/13
B 7/9
C 7/13
D 9/7
50 What is the tanQ?
A 9/13
B 7/9
C 7/13
D 9/7
The angle of slant of the Tower of Pisa is 84.3
A
B C
D
F
angle of slant
To find the trigonometric ratio of an angle, use a calculator or a trig table.
Check that your calculator is set for degrees (not radians) and round your answer to the ten thousandth place (4 decimal places).
Find the following:sin 84.3 = .9951cos 84.3 = .0993tan 84.3 = 10.0187
click
click
click
51 Evaluate sin 60. Round to the nearest ten
thousandth.
A 0.5
B 0.8660
C 1.7321
D 0.5774
52 Evaluate cos 60. Round to the nearest ten
thousandth.
A 0.5
B 0.8660
C 1.7321
D 0.5774
53 Evaluate tan 60. Round to the nearest ten
thousandth.
A 0.5
B 0.8660
C 1.7321
D 0.5774
Trig tables were used by early mathematicians and astronomers to calculate distances that they were unable to measure directly.
Today, calculators are usually used.
x
opp
adj
A A
A
54 Using B, which is the correct trig equation
needed to solve for x.
D
55
solve for x.
D
56
solve for x.
57 Using K, which is the correct trig equation needed to solve for x.
opp
adj
Using your calculator, find the tan 84.3Round your answer to 4 decimal places.
You can rewrite 10.0187 with a denominator of 1 and use the cross product property or multiply both sides of the equation by 5 using the multiplication property of equality (see next slide).
opp
adj
Multiply both sides of the equation by 5 using the multiplication property of equality.
Round your answer to the nearest hundredth. C
Round your answer to the nearest hundredth.
58 59 Find the length of LP. Round your answer to the nearest tenth.
Explain and use the relationship between the sine and cosine of complementary angles.
The sum of the interior angles of any triangle is equal to 180 degrees.
A and B are complementary angles. Complementary angles are two angles whose sum of their measures is 90 degrees.
The acute angles of a right triangle are always complementary.
60 For right triangle ABC, what is the measure of B?
A 30 degrees
B 50 degrees
C 60 degrees
D cannot be determined
61 If the , find the complementary angle?
A 20 degrees
B 70 degrees
C 160 degrees
D none of the above
In a right triangle, the acute angles are complementary. m A + m B = 90 53.1 + 36.9 = 90
sin A = 4/5 sin 53.1 = .7997cos B = 4/5 cos 36.9 = .7997sin A = cos Bsin 53.1 = cos 36.9The sine of an angle is equal to the cosine of its complement.
cos A = 3/5 cos 53.1 = .6004sin B = 3/5 sin 36.9 = .6004cos A = sin Bcos 53.1 = sin 36.9The cosine of an angle is equal to the sine of its complement.
sine function cosine function
Sine and Cosine are called co-functions of each other.Co-functions of complementary angles are equal.
62 Given that sin 10 = .1736, write the cosine of
a complementary angle.
A sin 10 = .1736
B sin 80 = .9848
C cos 10 = .9848
D cos 80 = .1736
63 Given that cos 50 = .6428, write the sine of
a complementary angle.
A sin 50 = .7660
B sin 40 = .6428
C cos 50 = .6428
D cos 40 = .7660
64 Given that cos 65 = .4226, write the sine of
a complementary angle.
A sin 25 = .4226
B cos 25 = .9063
C sin 65 = .9063
D cos 65 = .4226
65 What can you conclude about the sine and cosine of
45 degrees?
solve a right triangle means to find all 6 values in a triangle.
The and the
x
y
z
x
y
z
26
y
z
26
13.48
z
Try this...
D
11
C
x
y
z
C
x
y
z
You will need to use the
If sinθ = , θ = sin-1
If cosθ = , θ = cos-1
If tanθ = , θ = tan-1 θ
With the sine, cosine and tangent trig functions, if you know the angle θ and the measure of one leg, then you can find the measure of a leg of a triangle.
With the inverse sine, inverse cosine and inverse tangent trig functions, if you know the measures of 2 legs of a triangle, you can find the measure of the angle.
Pronounced inverse sine,inverse cosine,and inverse tangent.
θ = ( ) θ = θ = adjacent side
θ
The 3 Inverse Trigonometric Ratios
Remember:
66 67 the angle measure
68 the angle measure
C
θ
adj
hyp
A
θ
θ
A
θ
θ
A
θ
θ 69 Which is the correct trig equation to solve for
A
B
C
DD
70 Which is the correct trig equation to solve for
A
B
C
D D
71 Which is the correct trig equation to solve for
A
B
C
D
Try this... 72 Find CE.
C
D
73 Find m C.
C
D
74
C
D
75 76
77 Find the m P. 78 Find RT.
trigonometric ratios to solve word problems involving angles of elevation and depression?
79 How can you describe the angle relationship between
the angle of elevation and the angle of depression?
A corresponding angles
B alternate interior angles
C alternate exterior angles
D none of the above
Amy is flying a kite at an angle of 58
How high is the kite off the ground?
x
3 feet
sin
x = 134
Now, we must add in Amy's arm height.134 + 3 = 137
x
You are standing on a mountain that is 5306 feet high. You look do
.5774x = 5312
≈
Try this...
elevation is 55
If you are 5.5 feet tall, how far are you from the
base of the tree?
80
81 82
Law of Sines and Law of Cosines
The Law of Sines and Law of Cosines can be used to solve any triangle.
You can use the Law of Sines when you are given - 1. Two angle measures and any side length (AAS or ASA)2. Two side lengths and the measure of a non-included angle (SSA) when the angle is a right angle or an obtuse angle. The Law of Sines has a problem dealing with SSA when the angle is acute. There can be zero, one or two solutions.
Click on:
for more info.
You can use the Law of Cosines when you are given - 3. Three side lengths (SSS)4. Two side lengths and the measure of an included angle (SAS)
Law of Sines
sin A sinC a b c
To use the Law of Sines,
C
Given:
Prove:
ReasonsGivenDef of Altitude
then sin A sinC a b c
sin A sinC a b c
C
h
Draw an altitude from C to side AB
Let h be the length of the altitude
ABC with side lengths a, b, and c
Def of sine
Multiply by b. Mult Prop of =.
Substitution Prop of =
Divide by ab. Division Prop of =
Multiply by a. Mult Prop of =.
click
click
click
click
click
click
click
Given:
Prove:Reasons
Def of Altitude
sin A sinC a b c Draw an altitude from B to side AC
Let g be the length of the altitude
Def of sine
Multiply by c. Mult Prop of =.
Substitution Prop of =
Divide by ac. Division Prop of =
Multiply by a. Mult Prop of =.
g
C
Substitution Prop of =
click
click
click
click
click
click
click
C
h
C
Select the ratios based on the given information.
Given: m B, m C and BA (side c) (AAS)
Which ratios must be used first? Why?
sin A sinC a b c
C
First we can find the length side b.find the
C
Triangle Sum Theoremm A + m B + m C = 180o
C
sinA sinC a c
Now we find the length side a. Try this...
C
hint
Example...
C
83 Find the m A.
C
84 Which ratio must be used to find the length of
b or
C
85 What is the length of b?
C
86 What is the length of c?
C
C
If ABC has sides of length a, b, and c, then:
To use the Law Cosines, you must be given the sides(SSS)angle (SAS).
Law of Cosines
Given:
Prove:
ReasonsGiven
Def of Altitude
C
h
Draw an altitude CD from C to side AB. Let h be the length of the alt.
ABC with side lengths a, b, and c
Segment Addition Postulate
Multiply by b. Mult Prop of =.
Simplify
Substitution, equation (2)
C
Let x be the length of AD. Then (c-x) is the length of DB.
In ADC, cosA = x/b
x=b(cosA)
Definition of cosine
In ADC, Pythagorean Theorem
In CDB, Pythagorean Theorem
(1)
(2)
Associative Prop of Addition
Substitution, equation (1)
(similar reasoning shows that
)
click
click
click
click
click
click
click
click
click
click
click
C
The formula you choose depends on which angle you
To find the m A,
= 23 + 27256 = 529 + 729 - 1242(cosA)256 = 1258 - 1242(cosA)-1002 = -1242(cosA).8068 = cosAA = cos-1(.8068)
m A ≈ 36.22
C
= 16 + 27529 = 256 + 729 - 864(cosB)579 = 985 - 864(cosB)
.4699 = cosBB=cos-1(.4699)
C
36.22
or
Using 2 different methods, the answersare slightly different because of rounding.
To find the m C,
Use the Triangle Sum Theorem.
C
36.22
61.97
Try this...
C
87
8
9
C
88
8
9
C
89 Which formula would you use to find the m<A? 90
C
91 What is the m C?
C
92 What is the measure of B (ASA)?
B
A
C
93 The Law of Sines and Cosines is used to solve...
acute triangles
all triangles
Find the area of the triangle.
b is the base of the triangle b = 10.
h is the altitude (or height). It is the perpendicular
bisector of the base in an isosceles triangle.
Find the area of the triangle.
67.38
67.38
Given:
Prove:
ReasonsGiven
h
Let h be the length of the altitude
ABC with side lengths a, b, and c
Def of sine
Substitution Prop of =
Definition. Formula forthe area of a triangle.
Multiply by a. Mult Prop of =.
Commutative Prop ofMultiplication
94 Which of the following expressions can
be used to find the area of the triangle
below? Select all that apply.
A
B
C
D
F
95 Find the area of the triangle to the nearest tenth. 96 Find the area of the triangle to the nearest tenth.
97 Find the area of the triangle to the nearest tenth.
