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Difference between revisions of "2019 AMC 10B Problems/Problem 10"

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==Solution==
 
==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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Notice that whatever point we pick for <math>C</math>, <math>AB</math> will be the base of the triangle. Without loss of generality, let points <math>A</math> and <math>B</math> be <math>(0,0)</math> and <math>(0,10)</math>, since for any other combination of points, we can just rotate the plane to make them <math>(0,0)</math> and <math>(0,10)</math> under a new coordinate system. When we pick point <math>C</math>, we have to make sure that its <math>y</math>-coordinate is <math>\pm20</math>, because that's the only way the area of the triangle can be <math>100</math>.  
  
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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{\textbf{(A) }0}</math>.
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Now when the perimeter is minimized, by symmetry, we put <math>C</math> in the middle, at <math>(5, 20)</math>. We can easily see that <math>AC</math> and <math>BC</math> will both be <math>\sqrt{20^2+5^2} = \sqrt{425}</math>. The perimeter of this minimal triangle is <math>2\sqrt{425} + 10</math>, which is larger than <math>50</math>. Since the minimum perimeter is greater than <math>50</math>, there is no triangle that satisfies the condition, giving us <math>\boxed{\textbf{(A) }0}</math>.
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~IronicNinja
 
  
 
==Solution 2==
 
==Solution 2==
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WLOG let <math>AB</math> be a horizontal segment of length 10. Now, realize <math>C</math> has to lie on one of the lines parallel to <math>AB</math> and vertically 20 units away from it. But 10+20+20 is already 50, and this doesn't form a triangle. Otherwise, WLOG <math>AC<20</math>. Dropping altitude <math>CD</math> we have a right triangle <math>ACD</math> with hypotenuse <math>AC<20</math> and leg <math>CD=20</math> which is clearly impossible, giving us <math>\boxed{\textbf{(A) }0}</math>.
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Without loss of generality, let <math>AB</math> be a horizontal segment of length <math>10</math>. Now realize that <math>C</math> has to lie on one of the lines parallel to <math>AB</math> and vertically <math>20</math> units away from it. But <math>10+20+20</math> is already 50, and this doesn't form a triangle. Otherwise, without loss of generality, <math>AC<20</math>. Dropping altitude <math>CD</math>, we have a right triangle <math>ACD</math> with hypotenuse <math>AC<20</math> and leg <math>CD=20</math>, which is clearly impossible, again giving the answer as <math>\boxed{\textbf{(A) }0}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 22:11, 17 February 2019

The following problem is from both the 2019 AMC 10B #10 and 2019 AMC 12B #6, so both problems redirect to this page.

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?

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

Solution

Notice that whatever point we pick for $C$, $AB$ will be the base of the triangle. Without loss of generality, let points $A$ and $B$ be $(0,0)$ and $(0,10)$, since for any other combination of points, we can just rotate the plane to make them $(0,0)$ and $(0,10)$ under a new coordinate system. When we pick point $C$, we have to make sure that its $y$-coordinate is $\pm20$, because that's the only way the area of the triangle can be $100$.

Now when the perimeter is minimized, by symmetry, we put $C$ in the middle, at $(5, 20)$. We can easily see that $AC$ and $BC$ will both be $\sqrt{20^2+5^2} = \sqrt{425}$. The perimeter of this minimal triangle is $2\sqrt{425} + 10$, which is larger than $50$. Since the minimum perimeter is greater than $50$, there is no triangle that satisfies the condition, giving us $\boxed{\textbf{(A) }0}$.

Solution 2

Without loss of generality, let $AB$ be a horizontal segment of length $10$. Now realize that $C$ has to lie on one of the lines parallel to $AB$ and vertically $20$ units away from it. But $10+20+20$ is already 50, and this doesn't form a triangle. Otherwise, without loss of generality, $AC<20$. Dropping altitude $CD$, we have a right triangle $ACD$ with hypotenuse $AC<20$ and leg $CD=20$, which is clearly impossible, again giving the answer as $\boxed{\textbf{(A) }0}$.

See Also

2019 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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