CHAPTER 8
By:Fiona Coupe, Dani Frese, and Ale
Dumenigo
8-1 Similarity in right triangles
• Rt similarity- if the altitude is drawn to the hypotenuse of the triangle then the two small triangles are similar to each other
• Corollary 1- when the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse
• Corollary 2- when the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segments of the hypotenuse that is adjacent to that leg– Geometric mean- average in a geometric shape– Altitude- a line from a vertex of a triangle perpendicular to the
opposite side
C
BN
A
1. ACB ~ ANC by AA~Proportional sides
AB = ACAC AN AC is the geometric mean between AB and AN
2. ACB ~ CNB by AA~Proportional sides AB = BC BC NB BC is the geometric mean between AB and NB
3. ANC ~ CNBProportional sides AN = CN CN NB CN is the geometric mean between AN and NB
EXAMPLE FOR GEOMETIRC MEAN:
H
RJ
E
12
9 16
Find HJ, RE, RH and HE
RE= 9+16 RE = 25
HJ is the geometric mean between EJ and JR HJ = 9 = HJ HJ 16 HJ2 = 144 HJ = 12
RH is the geometric mean between RE and JR RH = 25 = RH RH 16 RH2 = 400 RH = 20
HE is the geometric mean between EJ and ER HE = 9 = HE HE 25 HE2 = 225 HE = 15
EXAMPLE FOR GEOMETRIC MEAN #2
Y
XA
Z
If XZ = 36, AX = 12, and ZY = 49 find ZA, YZ, YX
GEOMETRIC MEAN EXAMPLE #3
8.2 Pythagorean Theorem
If sides a and b are the legs of a right triangle and c is the hypotenuse then…
a2 + b2 = c2
Examples:1. A=2 B=3 and C=x 2. A=x B=x C=42²+3²= x² x² + x²= 164+9=x² 2x²=16√13=x 2x²/2= 16/2
x²=8x= √8 = 2√2
3. Find the diagonal of a rectangle with length 8 and width 48²+4²=c²64+16= c²80=c²√80=4√5=c
4
8
8.3 Converse of Pythagorean Theorem
• c²= a²+b² then the triangle is right• c²< a²+b² then the triangle is acute• c²> a²+b² then the triangle is obtuse
Examples:
• Sides: 6,7,8• Start by comparing the longest side to the
shorter ones• 8²= 64 • 6²+7²= 36+49 =85• 64 < 85• The triangle is acute
Common right triangles
• A=3 B=4 C=5• A=5 B=12 C=13• A=8 B=15 C=17• A=7 B=24 C=25
• These common right triangles also apply for their multiples
8-4 Special Right Triangles
• Theorem 8-6 - in a 45-45-90 triangle the hypotenuse is times as long as a leg
• Theorem 8-7- in a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg
45
45
30
60
A
A
A
A
2A
A
Find the missing sides for the two triangles
45
45
60
30
7
X
X
X = 7
12
A
BC
AC = 6CB = 6
EXAMPLES:
Solve for the missing sides
45
45
18
A
BC
30
60
9
X
YZ
TrigonometryTangent Ratio
• The word trigonometry comes from Greek words that mean “Triangle measurement.” • Tangent Ratio:
Tangent of <A = leg opposite <A
Opposite Leg
Adjacent LegA
leg adjacent <A
In abbreviated form: tan A = opposite adjacent
Example: Find tan X and tan Y.
Y
Z X
12 13
5
Tan X= leg opposite <X = 12 leg adjacent to <X = 5
Tan Y= leg opposite <Y = 5 leg adjacent to <Y = 12
Tangent ExampleExample: Find the value of Y to the nearest tenth
Y
56°32
Tan 56° = y 32
Solution: y= 32(tan 56°)Y= 32(1.4826)Y= 47.4432 or 47.4
Workout Problem:
Find x in this right triangle:
x
38°
46
The Sine and Cosine Ratio
• The ratios that relate the legs to the hypotenuse are the sine and cosine ratios.
Sine of <A= leg opposite <A hypotenuse
Cosine of <A= leg adjacent to <A hypotenuse
Adjacent Leg
Opposite Leg
A
hypotenuse
Find value of x and y to the nearest integer.
120
67 °
x
y
Sin 67 ° = x/120X= 120 sin 67 °∙X= 120(0.9205)X= 110.46 or 110
Cos 67 °= y/120Y= 120 cos 67 °∙Y= 120(0.3907)Y= 46.884 or 47
State 2 different equations you could use to find the value of x.
49 °
x
41 °
100
SOHCAHTOA
• SOH- sine (the angle measurement)= opposite leg/ hypotenuse
• CAH- Cosine (the angle measurement) = Adjacent leg/ hypotenuse
• TOA- Tangent (the angle measurement) = Opposite leg/Adjacent
Applications of Right Triangle Trigonometry
• The angle between the top horizontal and the line of sight is called an angle of depression. • An angle of elevation is the angle between the bottom horizontal and the line of sight.
Angle of elevation: 2°
Angle of depression 2°
Horizontal
x
If the top of the lighthouse is 25 m above sea level, the distance x between the boat and the base of the lighthouse can be found in 2 ways.
Tan 2° = 25/xX= 25/ tan 2°X= 25/0.0349X= 716.3
Tan 88°= x/25X= 25(tan 88°)X= 25(28.6363)X= 715.9
A good answer would be that the boat is roughly 700 m. from the lighthouse
Examples• A kite is flying at an angle of elevation about 40°. All 80 m of the string have been let out. Ignoring the sag in the string, find
the height of the kite to the nearest 10 m.
• Two buildings on opposite sides of a street are 40 m. apart. From the top of the taller building, which is 185 m high, the angle of depression to the top of the shorter building is 13°. Find the height of the shorter building.
40°
80x
Sin 40° = x/80
X= 51.4
185
13°40
x
185
Tan 13° = x/40X= 9.23185-9.23= 175.77