Triangles

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# Triangles - PowerPoint PPT Presentation

Triangles. Classifications of Triangles Sum of Angles in triangles Pythagorean Theorem Trig Ratios Area of Triangles. Triangle Review. Acute Triangle Obtuse Triangle Right Triangle Equilateral Triangle Isosceles Triangle Scalene Triangle Sum of the angles in a triangle is 180.

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## Triangles

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### Triangles

Classifications of Triangles

Sum of Angles in triangles

Pythagorean Theorem

Trig Ratios

Area of Triangles

Triangle Review
• Acute Triangle
• Obtuse Triangle
• Right Triangle
• Equilateral Triangle
• Isosceles Triangle
• Scalene Triangle
• Sum of the angles in a triangle is 180

5x

3x

10

10

3x

y+7

2x

55

10

3x=45 y+7=45

x=15 y=38

Isosceles Right Triangle

5x+55=180

5x=125

X=25

75,50,55

Scalene, Acute

5x=60

x=12

Equilateral, Acute

c

b

a

The Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

a2+ b2=c2

8m

8m

h

5

5

10m

Example 1: Find the Area

Because the triangle is isosceles,

the base is bisected.

Use pyth. Thm to find “h”

a2+b2=c2

52+h2=82

25+h2=64

h2=39

h=6.24

Area of Triangle

A=1/2 bh

A=1/2(10)(6.24)

A=31.2m2

=6.24

2nd base

65ft

65ft

3rd

base

Pitcher’s Mound

91.9ft

1st base

50ft

65ft

65ft

Home Plate

Example 2: In slow pitch softball, the distance between consecutive bases is 65 ft. The pitcher’s plate is located on the line between second based and home plate 50 ft from home plate. How far is the pitcher’s plate from second base? Justify your answer

You can usethe Pyth. Thm:

a2+b2=c2

652+652 = x2

4225+4225=x2

8450 = x 2

91.9ft from home plate to 2nd base = x

Total distance - PM to HP = 2nd to PM

91.9 - 50 = 41.9ft

Pythagorean Triple:

3 positive integers a,b,c, that satisfy a2+b2=c2

Example:

3,4,5 represent a Pythagorean Triple

32+42=52

9+16 = 25

25=25

B

c

a

A

C

b

Trig Ratios:

Sin A = Opposite Side

Hypotenuse

Hypotenuse

Tan A = Opposite Side

Hypotenuse

Opposite

B

Hypotenuse

5

3

A

C

4

Example 1: Find the ratio of the sin A, cos A and Tan A

Sin A = Opp

Hyp.

= 3

5

= 4

5

Hyp

Opposite

Tan A = Opp

= 3

4

B

5

3

A

C

4

Example 2: Find the ratio of the sin B, cos B and Tan B

Sin B = Opp

Hyp.

= 4

5

Hypotenuse

= 3

5

Hyp

Tan B = Opp

= 4

3

Opposite

Sin, Cos and Tan on your Calculator

Cos 13º = _______

.9744

.4540

Sin 27º = _______

2.2460

Tan 66º = _______

x

48º

100 ft

Example 4: Find the height of the silo.

Tan 48 = x

100

Solve by cross mult.

You can use Tan Ratio:

Tan A = opp

X = 100 ● tan 48

X = 111 ft.

x

38º

154 ft

Example 5: You are measuring the height of a tower. You stand 154 ft. from the base of the tower. You measure the angle of elevation from a point on the ground to the top of the tower to be 38º. Estimate the height of the tower.

Tan 38 = x

154

Tan A = opp

X = 154 ● tan 38

X = 120 ft.

14

9

47

x

x

15

x

22

35

19

52

Example 6: Other Variations: Solve for x

Sin 22 = x/14

14Sin22 = x

Cos 47 = 9/x

xCos 47 = 9

X = 9/cos47

Sin 35 = 15/x

xSin35 = 15

X = 15/Sin35

Cos 52 = x/19

19Cos 52 = x

x

B

10

8

C

A

x

Example 7:Solve the Right Triangle

Sides:

AB = 8

BC = 10

Missing AC: To find AC use pyth thm

82+102=x2

64+100 = x2

164 = x2

12.8 = x,

AC = 12.8

B

10

8

opp

C

A

x

Example 7:Solve the Right Triangle

Angles:

<B = 90

Missing <A and <C: To find find the missing angles, we will use INVERSE trig functions.

To get A by itself, we must do the opposite of Tan. This is called INVERSE TAN, it is Tan-1on your calculator

Tan A = 10

8

Tan A = opp

Tan A = 1.25

Tan-1Tan A = Tan-1 1.25

A = 51.34º

B

opp

10

8

C

A

x

Example 7:Solve the Right Triangle

Angles:

<B = 90

<A = 51.34

Missing <C: To find find the missing angles, we will use INVERSE trig functions.

To get A by itself, we must do the opposite of Tan. This is called INVERSE TAN, it is Tan-1on your calculator

Tan C = 8

10

Tan C = opp

Tan C = .8

Tan-1Tan C = Tan-1 .8

C = 38.66

B

10

8

C

A

x

Example 7:Solve the Right Triangle

Sides:

AB = 8

BC = 10

AC = 12.8

Angles:

m<B = 90

m<A = 51.34

m<C = 38.66