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Right Triangles
Chapter 8:
(page 284)
Similarity in Right Triangles
8-1:
(page 285)
If a, b, and x are positive numbers and a : x = x : b , then x is the
between a and b.
Notice that x is both
Example 1:
in the proportion.
Find the geometric mean between:
(a)
12 and 3
(b)
7 and 14
(d)
64 and 49
(e)
1 and 3
Theorem 8-1
(c)
(f)
If the altitude is drawn to the
5 and 20
100 and 6
of a right triangle,
then the two triangles formed are similar to the
triangle and to each other.
C
Given: ∆ ABC with Rt. ∠ACB and with altitude CN
Prove: ∆ ACB ~ ∆ ANC ~ ∆ CNB
A
N
N
A
B
N
C
C
Proof: All three triangles can be proven similar by the
B
.
Corollary 1
When the altitude is drawn to the
of a right triangle,
the length of the altitude is the
between the
segments of the hypotenuse.
Given:
!ABC with Rt.!ACB and with altitude CN
C
Prove:
AN CN
=
CN BN
A
Proof: From Theorem 8-1, ∆ ACN ~ ∆
N
B
, therefore, this can be proven because
corresponding sides of similar triangles are in
Corollary 2
.
When the altitude is drawn to the
of a right triangle,
each leg is the
between the hypotenuse and
the segment of the hypotenuse that is adjacent to that leg.
Given:
!ABC with Rt.!ACB and with altitude CN
C
Prove: (1)
AB AC
=
AC AN
(2)
AB BC
=
BC BN
Proof: (1) From Theorem 8-1, ∆ ABC ~ ∆
(2) From Theorem 8-1, ∆ ABC ~ ∆
A
N
B
, and
, therefore, (1) and (2) can be proven
because corresponding sides of similar triangles are in
.
Informal statements of the corollaries:
Y
X
A
Z
XA YA
=
YA AZ
piece of hypotenuse
altitude
=
altitude
other piece of hypotenuse
For leg XY :
XZ XY
=
XY XA
hypotenuse
leg
=
leg
piece of hypotenuse adjacent to leg
For leg YZ :
XZ YZ
=
YZ AZ
hypotenuse
leg
=
leg
piece of hypotenuse adjacent to leg
Corollary 1
Corollary 2
Example 2:
(a)
If CN=8 & NB=16, then AN =
C
A
N
B
.
(b)
If AN=8 & NB=12, then CN =
.
C
A
(c)
N
B
If AC=6 & AN=4, then NB =
.
C
A
(d)
N
B
If CB=20 & NB=16, then AB =
.
C
A
N
B
Assignment: Written Exercises, pages 288 & 289: 17-41 odd #’s
The Pythagorean Theorem
8-2:
(page 290)
One of the best known and most useful theorems in all of mathematics is the
Theorem. This was named after
, a Greek mathematician and philosopher.
Many proofs of this theorem exist, including one by President
.
Pythagorean Theorem
Theorem 8-2
In a right triangle, the
of the hypotenuse is equal to the
sum of the
of the legs.
C
Given: ∆ ABC ; ∠ACB is a right angle
Prove:
Proof:
A
Statements
B
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
ALSO: See #41 on page 289 and the Challenge Exercise on page 294.
Examples:
Find the value of “x”.
(1)
(2)
5
4
x
x
2 5
12
(3)
(4)
6
x
13
13
x
9
|----------- 10 ----------|
(5)
A
B
x
(6)
5
x
4 3
D
C
|----------------- 17 ------------------|
AC=12; BD=16
Assignment: Written Exercises, pages 292 & 293: 1-31 odd, 33-36
Prepare for Quiz on Lessons 8-1 & 8-2: Right Triangles
The Converse of the Pythagorean Theorem
8-3:
Theorem 8-3
If the
(page 295)
of one side of a triangle is equal to the sum of the
squares of the other two sides, then the triangle is a
triangle.
C
Given
∆ ABC with
Prove:
∆ ABC is a
triangle.
A
Example 1:
(a)
B
Are the given lengths sides of right triangles?
4, 7, 9
(b)
20, 21, 29
(c)
0.8, 1.5, 1.7
A triangle with sides of 3, 4, and 5 is a right triangle because
This is a very common triangle, called a
-
-
.
triangle.
Any triangle with sides 3n, 4n, and 5n , where n > 0, is also a right triangle, because
.
Multiples of any 3 lengths that form a right triangle will also form
triangles.
These groups of 3 lengths are called
.
Some Common Right Triangle Lengths:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
____________
____________
____________
____________
____________
____________
____________
____________
____________
____________
____________
____________
Theorem 8-4
If the square of the longest side of a triangle is less than the sum of the
squares of the other two sides, then the triangle is an
C
b
a
A
If
<
Theorem 8-5
triangle.
c
B
, then m∠C < 90º, and ∆ ABC is acute.
+
If the square of the longest side of a triangle is greater than the sum of the
squares of the other two sides, then the triangle is an
C
b
a
A
c
If
Example 2:
>
triangle.
B
, then m∠C > 90º, and ∆ ABC is obtuse.
+
If a triangle is formed with the given lengths, is it acute, right, or obtuse?
(1) 8, 12, 13
(2) 4, 4, 7
(3) 8, 9, 12
(4) 8, 11, 15
(5) 4, 5, 6
(6) 8, 9, 17
Assignment: Written Exercises, page 297: 1-17 odd #’s
Special Right Triangles
8-4:
Theorem 8-6
In a 45º-45º-90º ∆ , the hypotenuse is
(page 300)
times as long as a leg.
45º
_____
a
45º
a
Note: A 45º-45º-90º triangle is an isosceles right triangle with congruent
Example 1:
(a)
(b)
Find the length of the legs or the hypotenuse of each 45º-45º-90º triangle.
If the legs = 5,
then the hypotenuse =
(b)
.
If the legs = 5 6 ,
(d)
then hypotenuse =
(e)
.
.
If the hypotenuse = 10,
then the legs =
.
If the hypotenuse = 8 2 ,
then the legs =
(f)
.
If the legs = 3 2 ,
then the hypotenuse =
.
If the hypotenuse = 4 3 ,
then the legs =
.
In a 30º-60º-90º triangle,
Theorem 8-7
the hypotenuse is
the longer leg is
as long as the shorter leg, and
times as long as the shorter leg.
A
shorter leg =
30º
c
hypotenuse =
=
longer leg
=
b
=
60º
B
a
The shorter leg is opposite the
Example 2:
C
º angle and the longer leg is opposite the
º angle.
Using the side given, find the other 2 sides of each 30º-60º-90º triangle.
(a)
shorter leg = 2
hypotenuse =
longer leg =
(b)
hypotenuse = 12
shorter leg =
longer leg =
(c)
longer leg =
6
shorter leg =
hypotenuse =
Assignment: Written Exercises, pages 302 & 303: 1-19 odd #’s, 20-29 ALL #’s, BONUS: 32 & 36
Prepare for Quiz on Lessons 8-3 & 8-4
The Tangent Ratio
8-5:
(page 305)
Trigonometry, comes from 2 Greek words, which mean “
”.
Our study of trigonometry will be limited to
trigonometry.
The tangent ratio is the ratio of the lengths of the
.
B
A
C
Definition:
tangent of !A = tan A =
length of the leg opposite !A BC
= = ________
length of the leg adjacent to !A AC
tangent of !B = tan B =
length of the leg opposite !B AC
=
= ________
length of the leg adjacent to !B BC
opposite
tan =
! ___ ___ ___
adjacent
Example 1:
Express tan A and tan B as ratios.
B
(a)
tan A =
(b)
tan B =
17
C
15
A
The table on page 311 gives approximate decimal values of the tangent ratio for some angles.
Example 2:
(a) tan 20º ≈
(b) tan 87º ≈
“≈” means “is approximately equal to”
NOTE: Many calculators also give approximations of the tangent ratio.
The table can also be used to find an angle measure given a tangent value.
Example 3:
(a) tan
≈ .5774
(b) tan
≈ 4.0108
Many calculators use inverse keys to get values for angle measures for a tangent value.
Example 4:
Find the value of x to the nearest tenth and y to the nearest degree.
(a)
(b)
x
3
x
72º
37º
25
x≈
x≈
(c)
(d)
x
yº
5
5
89
yº
4
y≈
y≈
Assignment: Written Exercises, pages 308 & 309: 1-27 odd #’s
8-6:
The Sine and Cosine Ratios
Review :
The tangent ratio is the ratio of the lengths of the
(page 312)
.
The sine ratio and cosine ratio relate the legs to the
.
B
A
C
Definitions:
sine of !A = sin A =
length of the leg opposite !A BC
=
= ________
length of the hypotenuse
AB
length of the leg adjacent to !A AC
cosine of !A = cos A =
=
= ________
length of the hypotenuse
AB
Review :
tangent of !A = tan A =
length of the leg opposite !A BC
=
= ________
length of the leg adjacent to !A AC
A way to always remember the definitions (Note: these are NOT the definitions!):
sin =
opposite
hypotenuse
_______ _______ _______
cos =
adjacent
hypotenuse
_______ _______ _______
tan =
opposite
adjacent
_______ _______ _______
Express the sine and cosine of ∠ A & ∠ B as ratios.
Example 1:
B
(a)
sin A =
(b)
cos A =
13
(c)
sin B =
(d)
cos B =
C
Example 2:
A
Use the trigonometry table or a calculator to find the values.
(a)
sin 22º ≈
(c)
sin
Example 3:
12
≈ 0.8746
(b)
cos 79º ≈
(d)
cos
≈ 0.7771
Find the value of “x” & “y” to the nearest integer.
(a)
(b)
84
x
x
x
y
38º
y
55º
|------------- 14 --------------|
x ≈ ________ y ≈ ________
x ≈ ________ y ≈ ________
Example 4:
Find “n” to the nearest degree.
(a)
nº
14
20
n≈
(b)
Find the measures of the three angles of a 3-4-5 triangle.
Assignment: Written Exercises, pages 314 to 316: 1-23 odd #’s
Worksheet on Lessons 8-5 & 8-6: The Sine, Cosine, and Tangent Ratios
8-7:
Applications of Right Triangle Trigonometry
(page 317)
horizontal
------------------------------------------------------- A
1
2
B
The angles formed between the horizontal and the line of
are the:
ANGLE of DEPRESSION: when a point B is viewed from a higher point A, ie.
ANGLE of ELEVATION: when a point A is viewed from a lower point B, ie.
Examples 1:
At a certain time, a post 6 ft tall casts a 3 ft shadow. What is the angle of
elevation of the sun?
Examples 2:
A person in a lighthouse 22 m above sea level sights a buoy in the water. If the
angle of depression to the buoy is 25º, how far from the base of the lighthouse is
the buoy?
Assignment: Written Exercises, pages 318 to 320: 1-13 odd #’s
Worksheet on Lesson 8-7: Applications of Trigonometry
Prepare for Quiz on Lessons 8-5 to 8-7: Trigonometry
Prepare for Test on Chapter 8: Right Triangles
.