April 5th copyright2009merrydavidson

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Happy Birthday to:

Emily Besserer 9/4

Stephen Hodges 6/8

AREA FORMULAfor oblique triangles

AbcBacCabArea sin2

1sin

2

1sin

2

1

2;))()((

cbascsbsassArea

Why use one formula over the other?

Memorize these……..

Heron’s Formula:

1) Find the area for a triangular lot having two sides of lengths 90 meters and 52 meters and the

included angle of 1020.

a = b = c =

A = B = C =

90 52

102

Which formula should we use? Area = ½ ac sin B

Area = ½ (90)(52) sin 102 2289 metersarea

2) Find the area for a triangle having sides of 5, 7 and 10.

a = b = c =

A = B = C =

5 10 7

Which formula should we use?

2 16.25 unitsarea

2;))()((

cbascsbsassArea

11(11 5)(11 10)(11 7)Area

Review of Bearings

• We use bearings to describe the direction something is traveling or the direction in which we see something.

A bearing measures the acute angle that a path or line of sight makes with a fixed north/south line.

Reading Bearings

• Many directions are given using a combination of two cardinal headings and degree number

• North (N) and South (S) are always first• Then the number of degrees• Finally the East (E) or West (W) heading

• Example: N 53° W

W

N

E

S

AB

CD

O

40º

75º

25º

20º

Reading Bearings

Bearing to A: N 70° E

Bearing to B: N 40° W

Bearing to C: S 15° W

Bearing to D: S 25° E

Example 3A boat leaves the entrance of a harbor and travels 40 miles on a bearing of S64E. How many miles south and east did the has the boat traveled?

S64E40

miles

4064cos

a a

ba64cos40

a5.17

4064sin

b

b64sin40

b36

Alternate Bearing Method• Problems may just give you a single bearing

• 130 degrees

• This is always referenced from North clockwise to the bearing (heading).

N

130

Now application of oblique triangles.

3) Due to winds a tree grew so it was leaning 4 degrees from vertical. At a point 35 meters from the tree, the angle of elevation to the top of the tree is 23 degrees. Find the height?

3) Find the height?a = ? b = c = 35

A = 23 B = 94 C =

C

A BFind angle C

63

Use law of sines to find the height. 15.3 m

4) A plane flies 500 kilometers at a bearing of 316 degrees from Naples to Elgin. The plane then flies 720 kilometers to Canton (due west of Naples). Find the bearing of

the flight from Elgin to Canton?

Draw in little compass lines at the vertices.

44

46

44

Complementary angles gives you 460

Alternate interior angles gives you 440

C

B

A

Looking for this

Find angle B and subtract 440 for the bearing.

44

46

44

C

B

A

a = 500 b = c = 720

A = B = ? C = 46

Find angle A using law of sines

30 104 What is the bearing from Elgin to Canton?

S600W or 2400

3) A boat race runs along a triangular course marked with buoys A, B, and C. The race starts with boats headed west for 3700 meters. The other two sides of the course lie to the north of the first side and their lengths are 1700 meters and 3000 meters. Draw a visual picture of the course and then find the bearings for the final two legs of the race.

A

C

B3700 meters

3000 meters1700 meters

A

C

B3700 meters

3000 meters1700 meters

a = 1700 b = 3000 c = 3700

A = B = C =

Looking for this

C is the largest angle, find this

First. Store it!100.2 Now find angle B

using law of sines

52.9

What is the bearing from B to C? N 37.1o E

52.9

What is this angle?

What is the bearing from C to A?

S 63.1o E

HW: finish the WS