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

Mathcounts24 Feb 15, 2019

#1**+1 **

I had trouble with this question and didn't answer it at all on the AMC 10

Mathcounts24 Feb 15, 2019

#3**+2 **

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.

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!**

Melody Feb 15, 2019