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Diketahui P adalah titik berat ΔABC dan D adalah titik tengah BC. Jika B A = p ​ dan BC = q ​ maka P D = ....

Diketahui P adalah titik berat ΔABC dan D adalah titik tengah BC. Jika dan maka ....

  1. begin mathsize 14px style 1 over 6 p with rightwards arrow on top minus 1 third q with rightwards arrow on top end style

  2. begin mathsize 14px style negative 1 over 6 p with rightwards arrow on top plus 1 third q with rightwards arrow on top end style

  3. begin mathsize 14px style 1 over 6 p with rightwards arrow on top plus 1 over 6 q with rightwards arrow on top end style

  4. begin mathsize 14px style negative 1 third p with rightwards arrow on top plus 1 over 6 q with rightwards arrow on top end style

  5. begin mathsize 14px style 1 third p with rightwards arrow on top minus 1 over 6 q with rightwards arrow on top end style

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Pembahasan

Perhatikan gambar dari ΔABC berikut ini Misalkan E dan F masing-masing merupakan titik tengah AC dan AB. Karena D adalah titik tengah BC, maka AD, BE, dan CF merupakan garis berat ΔABC. Sehingga titik potongnya yaitu P merupakan titik berat ΔABC. Perhatikan bahwa Karena D adalah titik tengah BC, maka Kemudian karena P adalah titik berat ΔABC, maka BP : PE = 2 : 1. Sehingga Maka Sehingga

Perhatikan gambar dari ΔABC berikut ini

Misalkan E dan F masing-masing merupakan titik tengah AC dan AB.

Karena D adalah titik tengah BC, maka AD, BE, dan CF merupakan garis berat ΔABC. Sehingga titik potongnya yaitu P merupakan titik berat ΔABC.

Perhatikan bahwa

begin mathsize 14px style stack P D with rightwards arrow on top equals stack B D with rightwards arrow on top minus stack B P with rightwards arrow on top end style 

Karena D adalah titik tengah BC, maka begin mathsize 14px style stack B D with rightwards arrow on top equals 1 half stack B C with rightwards arrow on top equals 1 half q with rightwards arrow on top end style

 

Kemudian karena P adalah titik berat ΔABC, maka BP : PE = 2 : 1. Sehingga begin mathsize 14px style stack B P with rightwards arrow on top equals 2 over 3 stack B E with rightwards arrow on top end style

Maka

begin mathsize 14px style stack B P with rightwards arrow on top equals 2 over 3 stack B E with rightwards arrow on top equals 2 over 3 open parentheses stack B C with rightwards arrow on top plus stack C E with rightwards arrow on top close parentheses equals 2 over 3 open parentheses q with rightwards arrow on top plus 1 half stack C A with rightwards arrow on top close parentheses equals 2 over 3 open parentheses q with rightwards arrow on top plus 1 half open parentheses stack B A with rightwards arrow on top minus stack B C with rightwards arrow on top close parentheses close parentheses equals 2 over 3 open parentheses q with rightwards arrow on top plus 1 half open parentheses p with rightwards arrow on top minus q with rightwards arrow on top close parentheses close parentheses equals 2 over 3 open parentheses q with rightwards arrow on top plus 1 half p with rightwards arrow on top minus 1 half q with rightwards arrow on top close parentheses equals 2 over 3 open parentheses 1 half p with rightwards arrow on top plus 1 half q with rightwards arrow on top close parentheses equals 1 third p with rightwards arrow on top plus 1 third q with rightwards arrow on top end style  

Sehingga

begin mathsize 14px style stack P D with rightwards arrow on top equals stack B D with rightwards arrow on top minus stack B P with rightwards arrow on top equals 1 half q with rightwards arrow on top minus open parentheses 1 third p with rightwards arrow on top plus 1 third q with rightwards arrow on top close parentheses equals 1 half q with rightwards arrow on top minus 1 third p with rightwards arrow on top minus 1 third q with rightwards arrow on top equals negative 1 third p with rightwards arrow on top plus 1 half q with rightwards arrow on top minus 1 third q with rightwards arrow on top equals negative 1 third p with rightwards arrow on top plus 3 over 6 q with rightwards arrow on top minus 2 over 6 q with rightwards arrow on top equals negative 1 third p with rightwards arrow on top plus 1 over 6 q with rightwards arrow on top end style 

 

 

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