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MADAM MUNIRAH BINTI SHAMSUDDIN KOLEJ KEMAHIRAN TINGGI MARA, LENGGONG

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SITI MAISARAH BINTI MOHD AZIZI LG 2002755

TECHNICAL MATHEMATIC

OBLIQUE TRIANGLES (COSINE RULE) & AREA OF TRIANGLES

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INTRODUCTION

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This video lesson will show you how you can use the   Law of Cosines , a formula to help you find the missing side or angle of any kind of triangle.

In this video lesson, we are specifically looking at  Oblique Triangles to use the Law of Cosines and to finding the Area of Triangles.

Then, we will explore the steps to finding the Area of Triangles.

1.

2.

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OBLIQUE TRIANGLE

Any kind of triangle that is not a right triangle is an Oblique Triangle .

oblique triangle

Example

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LAW OF COSINES

In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of these two sides multiplied by the cosine of the included angle.

a 2  = b 2  + c  2  – 2bc.cos A b 2  = a 2  + c  2  – 2ac.cos B c 2  = a 2  + b  2  – 2ab.cos C

cos A  = b 2  + c 2   – a 2 2bc cos B  = a 2  + c 2  – b 2 2ac cos C  = a 2  + b 2  – c 2 2ab

OR

In reference to the triangle shown in Figure A, the   Law of Cosines , for any triangle ABC is :

Figure A

Cosine Rule is defined as:

The  Law of Cosine is also known as Cosine Rule .

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Three sides are given.

HOW TO USE COSINE RULES?

T wo sides and one included angle is given.

Cosine Rule can be used when the following values are given:

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Let’s look the examples…

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HOW TO USE COSINE RULES?

Figure 1 shows the triangle ABC. Given that ∡B = 68 ° , AB = 20.5 cm, BC = 12.2 cm. Find the length of AC.

Example 1: Given 2 sides and included angle.

Figure 1

Cosine Rule can be used when the following values are given:

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Solution 1

b2 = (20.5)2 + (12.2)2 - 680 = 408.04 + 148.84 + 187.38 = 369.5 = 19.22 cm

T wo sides and one included angle is given.

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HOW TO USE COSINE RULES?

Figure 2 shows the triangle PQR. Given that ∡R = 111.4 ° , PR = 20 cm, RQ = 25 cm. Find : a) Length PQ b) ∡ Q

Example 2: Given 2 sides and included angle.

Figure 2

Cosine Rule can be used when the following values are given:

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Solution 2

20 25 cm

a) b) r2 sin Q 20cm sin Q = 202 + 252 - cos 111.40 = 400 + 625 + 364.87 = 1389.87 = 37.28 cm sin 111.4 37.92cm 37.9cm = 0.4913 = 29.430

T wo sides and one included angle is given.

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HOW TO USE COSINE RULES?

Figure 2 shows the triangle ABC. Given that AB = 6.4 cm, BC = 4.7 cm and AC = 7.8 cm. Find : a) ∡A b) ∡C

Example 3: Given 3 sides.

Figure 3

Cosine Rule can be used when the following values are given:

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Solution 3

4.7 cm 6.4 cm 7.8 cm

sin LC b) 6.4 sin sin sin37.20 4.7 6.4sin37.020 4.7 = 0.8199 = 55.070

Three sides are given.

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What is Oblique Triangle ?

Now let’s continue our lesson to finding Area of Triangle …

How to use Law of Cosine or Cosine Rules ?

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What is Law of Cosine or Cosine Rules ?

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Figure B

Area of Triangle is an any space that is inside the borders of a triangle shape.

The measurement or size of a surface on a shape is called the area .

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Area of a Triangle 1

AREA OF TRIANGLE

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AREA OF TRIANGLE

Figure C

In reference to the Area of Triangle shown in Figure C, the formula for finding the Area of Triangle is:

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1 ax bx sin C 2 1 bxcx sin A 2 1 ax c x sin B 2

To find the area of a shape, we need to use a formula.

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FINDING THE AREA OF TRIANGLE

Figure 4

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13 cm 57.80 10 cm

Example 4

Solution 4

In Figure 4 shows Δ PQR. Such that PQ = 13 cm, QR = 10 cm and ∡ Q = 57.8 ° . Find the area of the triangle Δ PQR.

The area of the triangle PQR 1 x 13 x 10 sin 57.80 2 = 55.00 cm2

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FINDING THE AREA OF TRIANGLE

Figure 5

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Example 5

Solution 5

In figure 5 shows Δ ABC such that AC = 14 cm, BC = 11 cm and ∡ C = 125 ° . Find: AB ∡ B c) The area of the triangle ABC

1250

a) b) c) A 82 = (14)2+ = 196 + 121 + 176.7 A 82 sin ZB 14 sin XB = cos 1250 = 22.22 cm sin 1250 22.22 14xsin1250 22.22 = 31.10 Area of the triangle ABC 1 x 14 x 11 x sin 1250 2 = 63.1 cm2 = 0.5161

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LAW OF COSINE

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Using the Cosine Rule, solve triangle PQR in which p = 6.5 cm, q = 7.4 cm and ∠R = 58 °. What is the approximate length of side r ?

Question 1

6.5

A. r = 6.70 cm

A. r = 6.70 cm

B. r = 5.70 cm

B. r = 5.70 cm

C. r = 6.78 cm

C. r = 6.78 cm

D. r = 5.78 cm

D. r = 5.78 cm

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LAW OF COSINE

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Question 2

Use the information given in the figure to determine the value of c to the nearest tenth of a meter. a = 12.3 , b = 17.6.

A. 7.9 m

A. 7.9 m

B. 7.8 m

B. 7.8 m

C. 8.9 m

C. 8.9 m

D. 8.6 m

D. 8.6 m

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LAW OF COSINE

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Find the missing side.

Question 3

A. 15.9 in

A. 15.9 in

B. 19.2 in

B. 19.2 in

C. 521.2 in

C. 521.2 in

D. 22.8 in

D. 22.8 in

15 in. 1150 12 in.

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LAW OF COSINE

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Which Law would you use to find AB?

Question 4

A. Law of Sines

A. Law of Sines

B. Law of Cosine

B. Law of Cosine

C. Law of Gravity

C. Law of Gravity

D. Law of the Jungle

D. Law of the Jungle

26.3 92.20 27.2

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LAW OF COSINE

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Question 5

A. ∡ A = 27.9 °

A. ∡ A = 27.9 °

B. ∡ A = 37.8 °

B. ∡ A = 37.8 °

C. ∡ A = 37.02 °

C. ∡ A = 37.02 °

D. ∡ A = 18.6 °

D. ∡ A = 18.6 °

Solve the triangle ABC. Given that AB = 6.4 cm, BC = 4.7 cm and AC = 7.8 cm. Find ∡A.

4.7 cm 6.4 cm 7.8 cm

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LAW OF COSINE

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Question 6

A. 27.9 cm, 29 °

A. 27.9 cm, 29 °

B. 28 cm, 33.8 °

B. 28 cm, 33.8 °

C. 22.22 cm, 31.1 °

C. 22.22 cm, 31.1 °

D. 20.21 cm, 15.6 °

D. 20.21 cm, 15.6 °

In Δ ABC given that AC = 14 cm, BC = 11 cm and ∡ C = 125 ° . Find: AB ∡ B

1250

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Question 7

37 cm 1070 43 cm

Find: the area of the triangle PQR .

A. 220.76 cm 2

A. 220.76 cm 2

B. 560.34 cm 2

B. 560.34 cm 2

C. 760.74 cm 2

C. 760.74 cm 2

D. 64.41 cm 2

D. 64.41 cm 2

AREA OF TRIANGLE

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Question 8

Which a = 27.2 cm, b = 26.3 cm, and ∠ C = 92.2 °. Find the area of a triangle ABC.

A. 300.42 cm 2

A. 300.42 cm 2

B. 357.42 cm 2

B. 357.42 cm 2

C. 200.42 cm 2

C. 200.42 cm 2

D. 257.42 cm 2

D. 257.42 cm 2

26.3 92.20 27.2

AREA OF TRIANGLE

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What is the area of this triangle?

Question 9

A. 12 cm

A. 12 cm

B. 12 cm 2

B. 12 cm 2

C. 63.1 cm

C. 63.1 cm

D. 63.1 cm 2

D. 63.1 cm 2

1250

AREA OF TRIANGLE

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Which j = 6.9 cm, k = 9.7 cm, l = 14 cm and ∠ L = 113.93 °. What is the area of this triangle JKL ?

Question 10

A. 39.29 cm

A. 39.29 cm

B. 38 cm 2

B. 38 cm 2

C. 30 cm

C. 30 cm

D. 30.59 cm 2

D. 30.59 cm 2

14 cm 9.7 cm 6.9 cm

AREA OF TRIANGLE

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