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 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

Angles in a Triangles

1. Inner (Interior) Angles of Triangles

(Sudut Dalam Segitiga

A B

C

EXERCISES-1

2

3

4

6

5

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.

Investigate the exterior angles of triangles

72o

60o

48oA B

C

DE

F

Conclusion

1

2

3

1. Acute Triangle (Segitiga Lancip)

All the interior angles are acute angles

2. Obtuse Triangle (Segitiga Tumpul)

one of the interior angles is obtuse angle

3. Right Triangle (Segitiga Siku-Siku)

one of the interior angles is right angle

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

1. Scalene Triangle (Segitiga Sembarang)

AB

C

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

AB ≠ BC ≠AC

2. Isosceles Triangle (Segitiga Samakaki)

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

AB

C

3. Equilateral Triangle (Segitiga Samasisi)

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

AB = AC = BCA

B

C

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)

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

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)

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?

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)

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

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?

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)

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

• 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

• 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

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

Based on the following figure, find the value of x

Problem 2

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

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

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

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

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.

Problem 3

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

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 ....

Problem 5

The perimeter of triangle ABC below is … . C

8 cm 17 cm

D 6 cm A B

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

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

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

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

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

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

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

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

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

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

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

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.

Problem 420 cm

12 cm

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

A B

CD