एक समानांतर चतुर्भुज PQRS के तीन शीर्ष P(-3, 2, 3), Q(4, 3, -2), R(-5, 1, 0) हैं। तो चौथे शीर्ष S का निर्देशांक ज्ञात कीजिए। 

  1. (1, - 2, 3)
  2. (2, -3, 3)
  3. (3, -2, 3)
  4. (-12, 0, 5)

Answer (Detailed Solution Below)

Option 4 : (-12, 0, 5)
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संकल्पना:

 (\(\rm x_1, y_1, z_1\)) और (\(\rm x_2, y_2, z_2\)के मध्यबिंदु को निम्न द्वारा ज्ञात किया गया है, M = (\(\rm \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\))

 

गणना:

F11 Aman Kumar 1-4-2021 Swati D1

माना कि S(x, y, z) आवश्यक बिंदु है। तो विकर्ण QS का मध्यबिंदु

(\(\rm \frac{x+4}{2}, \frac{y+3}{2}, \frac{z-2}{2}\)) है। 

और PR का मध्यबिंदु निम्न है 

(\(\rm \frac{-3-5}{2},\frac{2+1}{2},\frac{3+0}{2}\)), अर्थात्, (\(\rm \frac {-8} 2 , \frac 3 2 ,\frac 3 2 \))

लेकिन समांनातर चतुर्भुज के विकर्णों के मध्यबिंदु एक-दूसरे से मेल खाते हैं। 

\(\rm \frac{x+4}{2}=​​\frac {-8} 2, \frac{y+3}{2} =\frac 3 2, \frac{z-2}{2}=\frac 3 2\)

इसलिए, x = -12, y = 0, और z = 5

आवश्यक बिंदु S(-12, 0, 5) है। 

अतः विकल्प (4) सही है। 

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