Question 1200868
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The vertices of triangle ABC are A(11,3), B(2k, k) and C(-1,-11).
(a) Find the two possible values of k if angle ABC is 90°.
(b) Draw the diagrams to show the two possible triangles.
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How the problem is posed/worded/formulated in the post, 
its educational meaning disappeared, in full.


Meanwhile, the meaning can be restored and problem can be nicely presented 
as a nice problem, if to re-formulate it in a right way.


This right way to re-formulate is as follows:


<pre>
    The vertices of triangle ABC are A(11,3), B(2k, k) and C(-1,-11).
    How to construct geometrically the point B (two possible positions)
    in a way that the triangle ABC be a right-angled triangle with the right angle at B ?
</pre>


Then the answer (the geometric construction) is as follows:


<pre>
    (a)  draw a circle with the diameter AC;

    (b)  then draw the straight line y = {{{(1/2)x}}}.

    (c)  Two intersection points of the line with the circle will give 
         two possible positions/locations of the vertex B.
</pre>