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