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Aim: What to do, What to do?!? So many formulas!! Where do we begin?. Do Now:. Regents Question How many distinct triangles can be forms if A = 30 o , side b = 12 and a = 8? 1) 1 2) 2 3) 3 4) 0. yi. 5i. 4i. vector: 2 + 3 i. 3i. (2 + 3 i ). 2i. (5 + 3 i ).

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Aim what to do what to do so many formulas where do we begin
Aim: What to do, What to do?!? So many formulas!! Where do we begin?

Do Now:

Regents Question

How many distinct triangles can be forms if A = 30o, side b = 12 and a = 8?

1) 1 2) 2 3) 3 4) 0


Adding complex numbers graphically

yi do we begin?

5i

4i

vector: 2 + 3i

3i

(2 + 3i)

2i

(5 + 3i)

(3 + 0i)

vector: 3 + 0i

i

x

1

-5

-4

-3

-2

-1

0

2

3

4

5

6

-i

-2i

-3i

-4i

-5i

-6i

Adding Complex Numbers Graphically

(2 + 3i) + (3 + 0i)

= (2 + 3) + (3i + 0i) =

= 5 + 3i

vector: 5 + 3i


Adding vectors

P do we begin?

S

resultant force

OS

O

R

Adding Vectors

Vector - a directed line segment that represents

directed force notation:

The vectors that represent the applied forces

form two adjacent sides of a parallelogram,

and the vector that represents the resultant

force is the diagonal of this parallelogram.


The laws

Law of Cosines: do we begin?

Law of Cosines:

Law of Sines:

Area of Triangle:

A = 1/2 ab sinC

The Laws!


General guidelines

Law of Cosines: do we begin?

Use the Law of Sines when the known information involves ASA, AAS, or SSA.

Law of Cosines:

Law of Sines:

Use the Law of Cosines when the known information involves SAS or SSS.

General Guidelines


Model problem

B do we begin?

A

350

105º40’

400

C

Model Problem

To determine the distance between 2 points, A and B, on opposite sides of a swampy region, a surveyor chose a point C that was 350 meters from point A and 400 meters from point B. If the measure of ACB was found to be 105º40’, find to the nearest meter, the distance, AB, across the swampy region.

Draw:

Given 2 sides & included angle:

Law of Cosines

Substitute

and solve:

AB = 598.414

AB2 = 358100

AB = 598 to nearest meter


Model problem1

A do we begin?

h

B

D

C

150’

Model Problem

A surveyor on the ground takes two readings of the angle of elevation of the top of a tower. From 150’ apart, the measures are 50o and 70o. Find the tower’s height to the nearest foot.

Find AD in ΔABD using Law of Sines;

then work in ΔADC for find AC.

70o

50o

AC  316’


Model problem2
Model Problem do we begin?

PA and PB are tangent to circle O at points A and B respectively. If PA = 10 cm and mP = 34o, find the length of chord AB to the nearest centimeter.

A

Tangents to a circle from an external point are congruent, making PB = 10.

With SAS known, use Law of Cosines.

10

P

34o

B

AB  6 cm.


Model problem3
Model Problem do we begin?

A canoe race is to be run over a triangular course marked by buoys A, B, and C. The distance between A and B is 100 yards, that between B and C is 160 yards, and that between C and A is 220 yards. Find to the nearest degree, the mABC.


Model problem4
Model Problem do we begin?

A diagonal of a parallelogram is 50 centimeters long and makes angles of 37o 10’ and 49o 20’, respectively, with the sides. Find the length of the shorter side of the parallelogram to the nearest centimeter.


Regents question 4 points
Regents Question – 4 points do we begin?

In triangle ABC, mA = 40 and mB = 56. The longest side of the triangle is 36 cm. Find the length of the shortest side to the nearest tenth of a centimeter.


Model problem5

h do we begin?

40o 10’

Model Problem

A vertical transmitting tower AB is located on a slope that is inclined 15o to the horizontal. At a point C, 80 feet down the slope from the foot of the tower, the tower subtends an angle of 40o 10’. Find to the nearest foot the height of the tower.

80’

15o


Regents question
Regents Question do we begin?

In triangle DEF, side e = 10, f = 8 and mD = 110. Find the length of the third side to the nearest tenth.

1) 218.7 2) 109.3 3) 14.8 4) 10.5


Model problem6
Model Problem do we begin?

A surveyor at point P sights two points X and Y that are on opposite sides of a lake. If P is 200 m. from X and 350 m. from Y, and mXPY = 40, find the distance from X to Y to the nearest meter.


Model problem7
Model Problem do we begin?

Some nylon fabric will be cut to cover the

kite frame shown below. Diagonal AC is

29 inches. What size should the angles be

at A, B, C, and D?

A

16 in.

D

16 in.

26 in.

B

26 in.

C


Model problem8

D do we begin?

C

110 lb.

B

A

Model Problem

Two forces act on a body at an angle of 72o, resulting in a force whose magnitude is 110 lb. If the magnitude of one of the original forces is 80 lb., find the magnitude of the other to the nearest pound.

Draw a parallelogram of forces.

80 lb.

80 lb.

Opposite sides are congruent: AD = BC = 80

72o

108o

?

Consecutive angles are supplementary: mABC = 108o

54 lb.

With SSA known in ΔABC, apply Law of Sines. To find AB, you must know mACB.


Regents questions 6 points
Regents Questions – 6 points do we begin?

The magnitude of the resultant of two forces acting on a body is 90 lbs. The angles between the forces and the resultant are 22030’ and 56o45’. Find the magnitude of the larger force to the nearest tenth of a pound.


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