Difference between revisions of "2019 AMC 10B Problems/Problem 10"
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<math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}</math> | ||
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+ | ==Solution== | ||
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+ | Notice that whatever point we pick for C, AB will be the base of the triangle. WLOG, points A and B are (0,0) and (0,10) [notice that for any other combination of points, we can just rotate the plane to be the same thing]. When we pick point C, we have to make sure the y value of C is 20, because that's the only way the area of the triangle can be 100. | ||
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+ | We figure that the one thing we need to test to see if there is such a triangle is when the perimeter is minimized, and the value of C is (x, 20). Thus, we put C in the middle, so point C is (5, 20). We can easily see that AC and BC will both be <math>\sqrt{20^2+5^2} \Rightarrow \sqrt{425}</math>. The perimeter of this minimized triangle is <math>2\sqrt{425} + 10</math>, which is larger than 50. Since the minimized perimeter is greater than 50, there is no triangle that satisfies the condition, giving us <math>\boxed{A) 0}</math> | ||
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+ | iron |
Revision as of 13:42, 14 February 2019
In a given plane, points and are units apart. How many points are there in the plane such that the perimeter of is units and the area of is square units?
Solution
Notice that whatever point we pick for C, AB will be the base of the triangle. WLOG, points A and B are (0,0) and (0,10) [notice that for any other combination of points, we can just rotate the plane to be the same thing]. When we pick point C, we have to make sure the y value of C is 20, because that's the only way the area of the triangle can be 100.
We figure that the one thing we need to test to see if there is such a triangle is when the perimeter is minimized, and the value of C is (x, 20). Thus, we put C in the middle, so point C is (5, 20). We can easily see that AC and BC will both be . The perimeter of this minimized triangle is , which is larger than 50. Since the minimized perimeter is greater than 50, there is no triangle that satisfies the condition, giving us
iron