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Jumat, 22 Maret 2013

LAYANG-LAYANG (KITES) DAN TRAPEZOID

A. LAYANG-LAYANG (KITES)

Layang -layang adalah segiempat yang dibentuk oleh dua segitiga sama kaki yang alasnya sama panjang dan berhimpit.

1. Sifat Layang-layang (Properties of Kites)
    a. Mempunyai dua pasang sisi sama panjang.
        AB = BC dan AD = CD
    b. Sepasang sudut yang berhadapan sama besar yakni sudut bagian sayap.
        Sudut BAD = Sudut BCD
    c. Mempunyai satu sumbu simetri yang merupakan diagonal terpanjang.
        Diagonal BD adalah sumbu simetri layang-layang.
    d. Diagonal bagian sayap membagi dua sama panjang dan tegak lurus terhadap diagonal lain.
        OA = OC = 1/2 AC  dan AC tegak lurus BD

2. Keliling dan Luas Layang-layang (Perimeter and Area of Kites)
    a. Keliling layang - layang = panjang keempat sisinya
                                           =   AB + BC + CD + AD

    b. Luas Layang-layang =     

                                                       

B. TRAPESIUM (TRAPEZOID)

Trapesium adalah segi empat yang mempunyai sepeasang sisi berhadapan sejajar.

                

1. Sifat - Sifat Trapesium (Properties of Trapezoid)
   a. Mempunyai sisi yang sejajar
       AB // DC tetapi AB tidak sama panjang dengan DC
   b. Mempunyai dua sudut lancip dan dua sudut tumpul
       Sudut Lancip = Sudut BAD dan Sudut ABC
       Sudut Tumpul = Sudut ADC dan Sudut BCD
   c. Jumlah sudut yang berdekatan di antara dua sisi sejajar ialah 180 derajat.
       Sudut A + Sudut D = 180
       Sudut B + Sudut C = 180

Berlaku khusus untuk trapesium sama kaki:
a. Mempunyai dua pasang sudut yang berdekatan sama besar.
b. Mempunyai dua diagonal yang sama panjang.

2. Keliling dan Luas Trapesium (Perimeter and Area of Trapezoid)
a. Keliling  Trapesium  = AB + BC + CD + AD

b. Luas Trapesium  

                                                          

SOAL-SOAL DAN PEMBAHASAN

1. Nyatakan masing-masing kalimat berikut ini BENAR atau SALAH untuk setiap layang-layang.
    a. Diagonal-diagonalnya sama panjang.
    b. Diagonal-diagonalnya saling berpotongan tegak lurus.
    c. Mempunyai dua pasang sisi yang sama panjang.
    d. Dua sudutnya yang berhadapan sama besar.

JAWAB:
    a. SALAH, diagonal -diagonal pada layang-layang TIDAK sama panjang.
    b.BENAR, diagonal-diagonal pada layang-layang berpotongan tegak lurus di O.
    c. BENAR, karena layang-layang dibentuk dari dua segitiga sama kaki.
    d. BENAR, sudut-sudut pada bagian sayap layang-layang sama besar.

2. Pada layang-layang ABCD berikut, besar sudut DAC  = 30 derajat dan Sudut CDB = 40 derajat.
    Tentukan:
    a. besar Sudut ADB;dan
    b. besar Sudut ABC.

JAWAB:
a. Sudut ADB = 180 - (Sudut AOD + Sudut DAC)
                      = 180 -  (90 + 30)
                      = 180 - 120
                      = 60
b. Sudut ABC = Sudut ADC
                       = Sudut ADB + Sudut CDB
                       = 60 + 40
                       = 100

3. Hitunglah luas layang-layang berikut.
JAWAB:
Diagonal 1 = 5 + 9 = 14 cm
Diagonal 2 = 4 + 4 = 8 cm
Luas Layang-layang  
                                
                                =  56 cm2

4. Luas suatu layang - layang adalah 60 cm2. Jika panjang salah satu diagonalnya 8 cm,
    hitunglah panjang diagonal yang lainnya.

JAWAB:
Luas layang-layang


120 = 8 x d1
d1   = 120 : 8
d1   = 15

Jadi, diagonal yang lain adalah 15 cm.

5. Diketahui luas persegi panjang ABCD pada gambar berikut 432 cm2.
    Luas daerah layang-layang adalah  . . .



JAWAB:
Luas persegi panjang = 3x . 2x
                        432   =
Luas layang - layang =                
                                     
                                  
                               
                               = 216 cm2

6. Pada trapesium sama kaki ABCD, panjang AD = BC, dan besar Sudut A = 65 derajat.
    a. Tentukan besar Sudut B!
    b. Tentukan besar Sudut C dan Sudut D!

JAWAB:
a. Sudut A = 65 derajat maka sudut B juga 65 derajat.
b. Sudut C berdekatan dengan sudut B maka sudut C = 180 - 65 = 115 derajat.
    Karena Sudut A = 65 derajat maka Sudut D pun 115 derajat.

7. Hitunglah luas trapesium berikut ini.
JAWAB:
(i) Luas Trapesium   
                                        
                               = 25 x 6
                              = 150 cm2

(ii) Luas Trapesium         
                           
                                       
                                = 22 x 15
                                = 330 cm2

8. ABCD adalah trapesium siku- siku di A dengan AB//CD.
    Jika besar sudut B : sudut C = 4 : 5,
    tentukan besar sudut-sudut trapesium itu!

JAWAB:
Sudut B dan Sudut C adalah sudut yang saling berdekatan dimana jumlah sudutnya = 180
Jumlah perbandingan adalah 4 + 5 = 9

 
        = 4 x 20  
        = 80

    
        = 5 x 20
        = 100

Jadi, Sudut A = 90, Sudut B = 80, Sudut C = 100 dan Sudut D = 90

9. Perbandingan panjang sisi- sisi sejajar sebuah trapesium adalah 4 : 3.
    Jika tinggi trapesium itu 8 cm, dan luasnya 84 cm2, hitunglah panjang sisi - sisi sejajar tersebut.

JAWAB:
Luas trapesium       
                            

                            
                     
                 a  + b = 21

Jumlah perbandingan sisi-sisi sejajar = 4 + 3 = 7

               
                a  = 4 x 3
                a = 12
           

b = 3 x 3
b = 9

Jadi sisi-sisi yang sejajar pada trapesium itu adalah 12 cm dan 9 cm.

10. Pak Ali membeli sebidang tanah dengan bentuk dan ukuran seperti pada gambar berikut.


Jika harga tanah Rp45.000,00/m2, maka Pak Ali  harus membayar sebesar . . . 

JAWAB:
Untuk menentukan tinggi kita gunakan TEOREMA PYTHAGORAS pada segitiga siku-siku.
Perhatikan gambar.

t  = 10

Luas tanah yang berbentuk trapesium     
                                                            
                                                            = 22 x 10
                                                            = 220 m2
Pak Ali harus membayar  = 220 x Rp45.000,-
                                       =Rp9.900.000,-

















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