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

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?

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

Source

Competency

Difficulty

Suggested Book

2019 AMC 10B Problem 10

Problem related to triangle

6 out of 10

Secrets in Inequalities.

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.

- https://www.cheenta.com/root-of-equation-bstat-hons-2005-problem-2/
- https://www.youtube.com/watch?v=GaQ5WtUgWJ4

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