TRIANGLE Segitiga

Triangle Around Us

6.1 Indentify the property of triangle based on their sides and angles BASE COMPETENCE 6.1 Indentify the property of triangle based on their sides and angles 6.2 Drawing a triangle, altitude, bisector, Median and axis on triangle 6.3 Count the perimeter and the area of triangle, and how to use in problem solving

Definition of triangle Triangle is a plane figure bounded by three non collinear lines and forming three inner (interior) angles. A b Side BC in front of angle A can be written as side a Side AC in front of angle B can be written as side b Side AB in front of angle C can be written as side c c C a B Triangle ABC Parts of Triangle: Points A, B adn C are called vertex AB,BC and AC are sides

Angles in a Triangles

1. Inner (Interior) Angles of Triangles (Sudut Dalam Segitiga The sum of inner angle is 180o C ∠A + ∠B + ∠C = 180o A B

EXERCISES-1

2

3

4 5 6

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 C 60o 48o B A 72o E D

Conclusion

1 2 3

A. Base on the size interior angle THE TYPES OF TRIANGLE A. Base on the size interior angle 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, 80 2. 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

determine the value of x and y and the type of triangle

B. Base on the lengths of their sides (Berdasarkan panjang sisinya) THE TYPES OF TRIANGLE B. Base on the lengths of their sides (Berdasarkan panjang sisinya) Scalene Triangle (Segitiga Sembarang) C A B Scalene Triangle is a triangle whose three sides are not equal in length. AB ≠ BC ≠AC

2. Isosceles Triangle (Segitiga Samakaki) B Isosceles Triangle is a triangle which the two sides are equal in length AB = AC

3. Equilateral Triangle (Segitiga Samasisi) C A B Equilateral Triangle is a triangle whose the three sides are equal in length AB = AC = BC

(Berdasarkan panjang sisi dan besar sudut) THE TYPES OF TRIANGLE Base on the lengths of their sides and the measure of its interior angle (Berdasarkan panjang sisi dan besar sudut) 1. Isosceles Triangle (Segitiga Samakaki) Obtuse angled-isosceles triangle (segitiga tumpul samakaki) Right angled-isosceles triangle (segitiga siku-siku samakaki Acute angled-isosceles triangle (segitiga lancip samakaki)

2. Equilateral Triangle (Segitiga samasisi) 60o 60o 60o Equilateral triangle (segitiga samasisi) It has equal measure of all sides and the measure of every interior angle = 60o

3. Scalene Triangle (Segitiga Sembarang) Obtuse angled-scalene triangle (segitiga tumpul sembarang) Right angled-scalene triangle (segitiga siku-siku sembarang) Acute angled-scalene triangle (segitiga lancip sembarang)

Answer the following questions Is there a right-angled equilateral triangle? (adakah segitiga siku-siku samasisi?) Is there a obtuse-angled equilateral triangle? (adakah segitiga tumpul samasisi?) Is there a acute-angled scalene triangle? (adakah segitiga lancip sembarang?) Is an equilateral triangle always an acute triangle?

Properties of Triangle 1. Isosceles Triangle (Segitiga Samakaki) It has 2 equal sides It has 2 equal angles It has one axis of simmetry (mempunyai satu sumbu simetri) It can fits its frame in 2 ways (dapat menempati bingkainya dengan 2 cara)

1. Isosceles Triangle (Segitiga Samakaki) C It can fits its frame in 2 ways (dapatmenempati bingkainya dengan 2 cara) A B D First posisition C B A D Second posisition

Properties of Triangle 2. Equilateral Triangle (Segitiga Samasisi) Answer the following questions: How many equal sides are in equilateral triangle? How many equal angles are in equilateral triangle? How many axis of simmetry are in equilateral triangle? In how many ways it can fits its frame?

2. Equilateral Triangle (Segitiga Samasisi) It has 3 equal sides It has 3 equal angles It has 3 axis of simmetry (mempunyai 3 sumbu simetri) It can fits its frame in 6 ways (dapat menempati bingkainya dengan 6 cara)

Properties of Triangle 3. Right Triangle (Segitiga Siku-siku) C Based on the figure on the left, then: BC2 = AC2 + AB2 Or a2 = b2 + c2 hypotenuse (sisi miring) kaki / sisi siku-siku leg / right side A leg / right side kaki / sisi siku-siku B For every right triangle: the square of its hypotenuse equals the sum of the square of the other sides. It called Pythagorean Theorem

a=17 cm 17 cm ? c=8 b=15 b2 + c2 = a2 Example 1 A triangle has legs measuring 8 cm and 15 cm, what is the length of the hypotenuse?? 1. Draw a picture 2. Write down Pythagorean theorem a=17 cm ? c=8 b2 + c2 = a2 3. Substitute in what you know b=15 152+ 82 =a2 225+ 64 = a2 4. Take square root!! 289= a2 17 cm 17= a

b2 + c2 = a2 Example 2 b=12 a=20 ? C=16 cm 122+ c2 = 202 144+ c2 = 400 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 picture 2. Write down Pythagorean theorem b=12 a=20 b2 + c2 = a2 3. Substitute in what you know C=16 cm ? 122+ c2 = 202 4. Take square root!! 144+ c2 = 400 c = √256 = 16 cm c2= 400 - 144 c2= 256

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

Problem 2 Based on the following figure, find the value of x

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

Bilangan Tripple Pythagoras Yaitu: 3 bilangan yang dapat digunakan sebagai sisi-sisi dari suatu segitiga siku-siku Dasar Kelipatan Sisi miring Dua sisi siku-siku 5 3 4 2,5 10 15 1,5 6 9 2 8 12 13 6,5 26 24 17 8,5 34 7,5 30 16 29 21 20 58 42 40 25 7 50 48 14

Perimeter of Triangle (Keliling Segitiga) C Perimeter of ∆ABC = AB + BC + AC = c + b + a = a + b + c a cm b cm A c cm B

Problem 1 In an isosceles triangle ABC. AB=BC, if AB = 15 cm and AC = 10 cm. Find the perimeter of triangle ABC Perimeter = AB + BC + AC = 15 + 15 +10 = 40 cm B 15 cm 15 cm A 10 cm C

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 4 Observe the following figure. If the perimeter of triangle PQR on the left = 180 cm, so the length of QR is .... 2x 3x Q 4x R

Problem 5 The perimeter of triangle ABC below is … . C 8 cm 17 cm D 6 cm A B

Area of Triangle Area of a triangle = ½ (base x altitude) = ½ b.h C AB is the base of ∆ABC, DC is altitude/height of ∆ABC A D B Area of a triangle = ½ (base x altitude) = ½ b.h

Example: R If PQ = 10 cm, RS= 12 cm and PR = 13 cm. The area of ∆PQR is…. S Q P Solution The 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 2 C Find the area of the green region, if: AB = 20 cm DC = 6 cm DE = 5 cm D A E B Solution: Area of green = Area ∆ABC – area ∆ABD = ½ . AB. CE – ½ .AB. DE = ½.20.11 – ½.20.5 = 60cm2

Problem 3 R Find the area of the blue region, if: PQ = 10 cm RS = 12 cm TU = 4 cm T P U Q S Solution: Area of Blue = Area ∆PQR – area ∆PQT = ½ . PQ. RS – ½ .PQ. TU = ½.10.12 – ½.10.4 = 60 – 20 = 40 cm2

Problem 4 The 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. 4x 150 = 6x2 25 = x2 5 = x 4x 3x The right sides are : 15 cm and 20 cm. Using pythagorean theorem, the length of hypotenuse = 25 cm Perimeter = 15 cm + 20 cm + 25 cm = 60 cm

Base and Altitude of Triangle 1. Right Triangle C Remember! Altitude is perpendicular to the base (tinggi segitiga tegak lurus dengan alas) D A B AC is the altitude to the base AB Area of ∆ABC = ½ x AB x AC AD is the altitude to the base BC Area of ∆ABC = ½ x AD x BC

Example 1 If AB = 8 cm and AC = 6 cm, then find: Length of BC Area of ∆ABC Length of AD C D 6 cm A 8 cm B

Base and Altitude of Triangle 2. Acute Triangle C DC is the altitude to the base AB Area of ∆ABC = ½ x AB x DC E F AE is the altitude to the base BC Area of ∆ABC = ½ x BC x AE A D B FB is the altitude to the base AC Area of ∆ABC = ½ x AC x FB

Example 2: If PQ = 16 cm, RQ = 15 cm and PT = 8 cm. Find the area of ∆PQR. Find the length of RS. R T P S Q

Base and Altitude of Triangle 3. Obtuse Triangle C DC is the altitude to the base AB Area of ∆ABC = ½ x AB x DC E AE is the altitude to the base BC Area of ∆ABC = ½ x BC x AE B A D F FB is the altitude to the base AC Area of ∆ABC = ½ x AC x FB

Example 3 If HD = 9 cm, DE = 7 cm and FH = 12 cm. Find the area of ∆DEF. Find the length of DG. F G H D E

Problem 4 D 20 cm C 12 cm A B ABCD is a rectangle (persegi panjang), then find the area of the shaded region.