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Let B = (20, 14) and C = (18, 0) be two points in the plane. For every line L passing through B, we
color red the foot of the perpendicular from C to L. The set of red points enclose a bounded region of
area A. Find greatest integer not exceeding A.

Guest Oct 26, 2014

Best Answer 

 #4
avatar+93301 
+5

I haven't looked at this yet but CPhill was very impressed with anon's logic.

I am very impressed with CPhill's pic.

I shall have to work out what it all means.     

Melody  Oct 27, 2014
 #1
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+5

the region is a circle, diameter is BC. the area is pi*BC*BC/4 = pi*200/4 < 157.

157 is the answer.

Guest Oct 26, 2014
 #2
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0

Thanks!

Guest Oct 26, 2014
 #3
avatar+88839 
+5

This one is interesting....it may be a little difficult to picture what is going on....here is a diagram.....

 

Note that we have a set of lines drawn through B. And, drawing a line from C to each line through B such that they are perpendicular at their intersection generates a set of right triangles. And, as Anonymous has pointed out, the locus of all possible points at the perpendiculars generates a circle with a radius of BC/2. And if we let BC = the distance from (20,14) to (18,0), then BC = √200. And the area of the circle is pi*(√200/2)2  = pi*(200/4) = 50pi = 157.07. And the greatest integer not exceeding this is 157.

Thank you for this great answer, Anonymous......I couldn't picture what was going on until I read your response and "sussed it out" for myself with a picture!!!

 

CPhill  Oct 26, 2014
 #4
avatar+93301 
+5
Best Answer

I haven't looked at this yet but CPhill was very impressed with anon's logic.

I am very impressed with CPhill's pic.

I shall have to work out what it all means.     

Melody  Oct 27, 2014

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