Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of...

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Segitiga TRIANGLE

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Transcript of Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of...

Page 1: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Segitiga

TRIANGLE

Page 2: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Triangle Around Us

Page 3: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.
Page 4: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Definition of triangle

A

B

C

Triangle ABC

a

b

c

Parts of Triangle:Points A, B adn C are called vertexAB,BC and AC are sides

Triangle is a plane figure bounded by three non collinear lines and forming three inner (interior) angles.

Side BC in front of angle A can be written as side aSide AC in front of angle B can be written as side bSide AB in front of angle C can be written as side c

Page 5: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Angles in a Triangles

Page 6: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1. Inner (Interior) Angles of Triangles

(Sudut Dalam Segitiga

A B

C

Page 7: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

EXERCISES-1

Page 8: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2

Page 9: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

3

Page 10: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

4

6

5

Page 11: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2. Exterior Angles of Triangle

In Exercise 1 you were dealing with the angles inside a triangle, called interior angles. In

this section we will look at the angles outside a triangle, called exterior angles.

If one side of a triangle is extended, the angle between this extension and the

triangle is called an exterior angle.

Page 12: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Investigate the exterior angles of triangles

72o

60o

48oA B

C

DE

F

Page 13: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Conclusion

Page 14: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1

2

3

Page 15: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1. Acute Triangle (Segitiga Lancip)

All the interior angles are acute angles

Page 16: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2. Obtuse Triangle (Segitiga Tumpul)

one of the interior angles is obtuse angle

Page 17: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

3. Right Triangle (Segitiga Siku-Siku)

one of the interior angles is right angle

Page 18: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Exercise• Determine the type of triangle bellow if

1. The angle are : 65, 35, 802. The angle are : 25, 60, 95

3. The angle are : 54, 56, 70 4. Two angle are : 73, 34, 5. The ratio of angle is 3 : 4 : 5 6. The ratio of angle is 2 : 3 : 4 7. The angle is 6x, 2x + 3, 4x +9

Page 19: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.
Page 20: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1. Scalene Triangle (Segitiga Sembarang)

AB

C

Scalene Triangle is a triangle whose three sides are not equal in length.

AB ≠ BC ≠AC

Page 21: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2. Isosceles Triangle (Segitiga Samakaki)

Isosceles Triangle is a triangle which the two sides are equal in lengthAB = AC

AB

C

Page 22: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

3. Equilateral Triangle (Segitiga Samasisi)

Equilateral Triangle is a triangle whose the three sides are equal in length

AB = AC = BCA

B

C

Page 23: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1. Isosceles Triangle (Segitiga Samakaki)

Right angled-isosceles triangle(segitiga siku-siku samakaki

Acute angled-isosceles triangle(segitiga lancip samakaki)

Obtuse angled-isosceles triangle(segitiga tumpul samakaki)

Page 24: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2. Equilateral Triangle (Segitiga samasisi)

Equilateral triangle(segitiga samasisi)

It has equal measure of all sides and the measure of every interior angle = 60o

60o 60o

60o

Page 25: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

3. Scalene Triangle (Segitiga Sembarang)

Right angled-scalene triangle(segitiga siku-siku sembarang)

Acute angled-scalene triangle(segitiga lancip sembarang)

Obtuse angled-scalene triangle(segitiga tumpul sembarang)

Page 26: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Answer the following questions

1. Is there a right-angled equilateral triangle? (adakah segitiga siku-siku samasisi?)

2. Is there a obtuse-angled equilateral triangle? (adakah segitiga tumpul samasisi?)

3. Is there a acute-angled scalene triangle? (adakah segitiga lancip sembarang?)

4. Is an equilateral triangle always an acute triangle?

Page 27: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Properties of Triangle1. Isosceles Triangle (Segitiga Samakaki)

a. It has 2 equal sidesb. It has 2 equal anglesc. It has one axis of simmetry

(mempunyai satu sumbu simetri)d. It can fits its frame in 2 ways (dapat

menempati bingkainya dengan 2 cara)

Page 28: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

1. Isosceles Triangle (Segitiga Samakaki)

It can fits its frame in 2 ways (dapatmenempati bingkainya dengan 2 cara)

A B

C

DFirst posisition

B A

C

DSecond posisition

Page 29: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Properties of Triangle2. Equilateral Triangle (Segitiga Samasisi)

Answer the following questions:a. How many equal sides are in

equilateral triangle?b. How many equal angles are in

equilateral triangle?c. How many axis of simmetry are in

equilateral triangle?d. In how many ways it can fits its frame?

Page 30: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

2. Equilateral Triangle (Segitiga Samasisi)

a. It has 3 equal sidesb. It has 3 equal anglesc. It has 3 axis of simmetry (mempunyai 3

sumbu simetri)d. It can fits its frame in 6 ways (dapat

menempati bingkainya dengan 6 cara)

Page 31: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Properties of Triangle3. Right Triangle (Segitiga Siku-siku)

hypotenuse(sisi miring)

BA

C

leg / right sidekaki / sisi siku-siku

leg

/ ri

ght

sid

eka

ki /

sisi

sik

u-si

ku

For every right triangle:the square of its hypotenuse equals the sum of the square of

the other sides.It called Pythagorean Theorem

Based on the figure on the left, then:BC2 = AC2 + AB2

Ora2 = b2 + c2

Page 32: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

• A triangle has legs measuring 8 cm and 15 cm, what is the length of the hypotenuse??

1. Draw a picture2. Write down Pythagorean

theorem c=8

b=15

?b2 + c2 = a2

3. Substitute in what you know

152+ 82 =a2

225+ 64 = a2

289= a2

4. Take square root!!

17= a 17 cm

Example 1

a=17 cm

Page 33: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

• A triangle has hypotenuse measuring of 20 cm and of its legs measuring 12 cm. Find the length of the other legs

1. Draw a picture2. Write down Pythagorean

theoremb=12 a=20

?

b2 + c2 = a2

3. Substitute in what you know

122+ c2 = 202

144+ c2 = 400c2= 400 - 144

4. Take square root!!

c2= 256

c = √256 = 16 cm

Example 2

C=16 cm

Page 34: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 1Based on the following figure, form the equation using Pythagorean Theorem

Page 35: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Based on the following figure, find the value of x

Problem 2

Page 36: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 3Based on the following figure, find the value of y

Page 37: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Bilangan Tripple PythagorasYaitu: 3 bilangan yang dapat digunakan sebagai sisi-sisi dari suatu segitiga siku-siku

Dasar Kelipatan

Sisi miring Dua sisi siku-siku Sisi miring Dua sisi siku-siku

5 3 4 2,51015

1,569

2812

13 12 5 6,526

624

2,510

17 15 8 8,534

7,530

416

29 21 20 58 42 40

25 24 7 50 48 14

Page 38: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Perimeter of Triangle (Keliling Segitiga)

A B

C Perimeter of ∆ABC = AB + BC + AC

= c + b + a= a + b + c

a cm

c cm

b cm

Page 39: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 1In an isosceles triangle ABC. AB=BC, if AB = 15 cm and AC = 10 cm. Find the perimeter of triangle ABC

A

B

15 cm

10 cm

15 c

m

C

Perimeter = AB + BC + AC

= 15 + 15 +10= 40 cm

Page 40: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 2

The perimeter of triangle ABC is 120 cm. If AB:BC:AC=3:4:5, the length of AB is….Solution:AB + BC + AC = 120 cm

AB = (3/12) x 120 cm = 30 cm.

Page 41: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 3

The perimeter of triangle ABC is 84 cm. If a : b : c = 5 : 3 : 4, the length of BC is….

Page 42: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 4Observe the following figure.

Q

P

3x

4x

2x

R

If the perimeter of triangle PQR on the left = 180 cm, so the length of QR is ....

Page 43: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 5

The perimeter of triangle ABC below is … . C

8 cm 17 cm

D 6 cm A B

Page 44: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Area of Triangle

A

C

BD

AB is the base of ∆ABC,DC is altitude/height of ∆ABC

Area of a triangle = ½ (base x altitude)

= ½ b.h

Page 45: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Example:

P Q

R

S

If PQ = 10 cm, RS= 12 cm and PR = 13 cm.The area of ∆PQR is….

SolutionThe area of ∆PQR = ½ x base. height

= ½ x PQ.RS= ½ x 10 x 12= 60 cm2

Page 46: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 1

The ratio of base and altitude of a triangle is 4 : 5. If the area of the triangle is 90 cm2, then find the base and altitude of the triangle.

Solution:Area = ½ .b. h

90 = ½ . 4x. 5x

90 = 10x2

9 = x2

3 = x

5x

4x

Page 47: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 2Find the area of the green region, if:AB = 20 cmDC = 6 cmDE = 5 cmA B

C

D

E

Solution:Area of green = Area ∆ABC – area ∆ABD

= ½ . AB. CE – ½ .AB. DE= ½.20.11 – ½.20.5 =

60cm2

Page 48: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 3Find the area of the blue region, if:PQ = 10 cmRS = 12 cmTU = 4 cmP Q

R

S

T

USolution:Area of Blue = Area ∆PQR – area ∆PQT

= ½ . PQ. RS – ½ .PQ. TU= ½.10.12 – ½.10.4 = 60 – 20 = 40 cm2

Page 49: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 4The ratio of right sides of a right triangle is 3 : 4. if the area of the triangle is 150 cm2, find the perimeter of the triangle.

Solution:Area = ½ .b. h

150 = ½ . 3x. 4x150 = 6x2

25 = x2

5 = x 3x

4x

The right sides are : 15 cm and 20 cm. Using pythagorean theorem, the length of hypotenuse = 25 cmPerimeter = 15 cm + 20 cm + 25 cm = 60 cm

Page 50: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Base and Altitude of Triangle

1. Right Triangle

A B

C

D

Remember!Altitude is perpendicular to the base(tinggi segitiga tegak lurus dengan alas)

AC is the altitude to the base ABArea of ∆ABC = ½ x AB x AC

AD is the altitude to the base BCArea of ∆ABC = ½ x AD x BC

Page 51: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Example 1

A B

C

D

6 cm

8 cm

If AB = 8 cm and AC = 6 cm, then find:a. Length of BCb. Area of ∆ABCc. Length of AD

Page 52: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Base and Altitude of Triangle2. Acute Triangle

A B

C

D

DC is the altitude to the base ABArea of ∆ABC = ½ x AB x DC

AE is the altitude to the base

BCArea of ∆ABC = ½ x

BC x AE

EF

FB is the altitude to the base ACArea of ∆ABC = ½ x AC x FB

Page 53: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Example 2:

If PQ = 16 cm, RQ = 15 cm and PT = 8 cm.a. Find the area of ∆PQR.b. Find the length of RS.

P Q

R

S

T

Page 54: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Base and Altitude of Triangle3. Obtuse Triangle

A B

C

D

DC is the altitude to the base ABArea of ∆ABC = ½ x AB x DC

AE is the altitude to the base

BCArea of ∆ABC = ½ x

BC x AE

E

F

FB is the altitude to the base ACArea of ∆ABC = ½ x AC x FB

Page 55: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Example 3

D E

F

H

G

If HD = 9 cm, DE = 7 cm and FH = 12 cm.a. Find the area of ∆DEF.b. Find the length of DG.

Page 56: Segitiga TRIANGLE. Triangle Around Us Definition of triangle A B C Triangle ABC a b c Parts of Triangle: Points A, B adn C are called vertex AB,BC.

Problem 420 cm

12 cm

ABCD is a rectangle (persegi panjang), then find the area of the shaded region.

A B

CD


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Triangle abc a²a² b²b² c²c² Blue*5121325144169 Green78134964169 Orange*34591625 Pink656362536 Purple*68103664100 White*7242549576625 Yellow*8151764225289.

Triangle abc a²a² b²b² c²c² Blue*5121325144169 Green78134964169 Orange*34591625 Pink656362536 Purple*68103664100 White*7242549576625 Yellow*8151764225289.