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I kinda gave up on the last problem... so new problem :)

 

Let  \(a = 4 + 3i,\) \(b = 1 -2i,\)and \(c = 8 - 5i.\)  The complex number d is such that a, b, c, and d form the vertices of a parallelogram, when plotted in the complex plane. Enter all possible values of d, separated by commas.

 Jun 20, 2019
edited by Pushy  Jun 20, 2019
edited by Pushy  Jun 20, 2019
edited by Pushy  Jun 20, 2019
 #1
avatar+8963 
+3

I found three different possible values of  d  using this graph:

 

https://www.desmos.com/calculator/cwbrobjoav

 

You can turn on the different parallelograms by clicking the circle next to the name.

Here's a picture with all of them turned on:

 

The different possible values of  d  are:

 

 

-3 + 6i

 

5 - 10i

 

 

11 + 0i 
 Jun 21, 2019
 #2
avatar+25344 
+3

Let \(\large{a = 4 + 3i}\), \(\large{b = 1 -2i}\), and \(\large{c = 8 - 5i}\).  
The complex number d is such that a, b, c, and d form the vertices of a parallelogram,
when plotted in the complex plane. Enter all possible values of d, separated by commas.

 

\(\begin{array}{|rclclcl|} \hline \mathbf{d} &=& -a+b+c &=& -(4+3i)+ (1 -2i) + (8 - 5i) &=& \mathbf{5-10i} \\\\ \mathbf{d} &=& a-b+c &=& (4+3i)- (1 -2i) + (8 - 5i) &=& \mathbf{11+0i} \\\\ \mathbf{d} &=& a+b-c &=& (4+3i)+ (1 -2i) - (8 - 5i) &=& \mathbf{-3+6i} \\ \hline \end{array} \)

 

laugh

 Jun 21, 2019

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