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The given problem is related to the calculation of area of triangle and distance between two points.

## Try the problem

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 $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?

$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

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2019 AMC 10B Problem 10

Problem related to triangle

6 out of 10

Secrets in Inequalities.

## Use some hints

First hint

Notice that it does not matter where the triangle is in the 2D plane so for our easy access we can select two points A and B in any place of choice.

Second Hint

So we can actually select any two points A and B such that they are 10 units apart so lets the points are $A(0,0)$ and $B(10,0)$ , as they are 10 units apart.

Final Step

Now we can select the point C such that the perimeter of the triangle is 50 units. and then we can apply the formula of area to calculate the possible positions of C.