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10. In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of ABC is 50 units and the area of ABC is 100 units?

 

A) 0

 

B) 2

 

C) 4

 

D) 8

 

E) Infinitely Many

 Feb 15, 2019
 #1
avatar+53 
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I had trouble with this question and didn't answer it at all on the AMC 10

 Feb 15, 2019
 #2
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I got 0 for some reason.... but im probably wrong.

 Feb 15, 2019
 #3
avatar+99301 
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In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of ABC is 50 units and the area of ABC is 100 units?

 

Without much thought I would assume the answer is infinite.

But let's take a look.

I am using the triangle below. You can see that the sides add up to 50.

 

Now

\(Area= 0.5*10*h=100\\ 5*h=100\\ h=20\\ \)

I just did a stack of work that i will not even bother to display.  angry

 

BECAUSE

 

If the perpendicular height must be 20 then the sum of the two slant heights of the triangle must be more than 40.

But this is impossible so there are no solutions.    I think that you guessed correctly!

 

 Feb 15, 2019

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