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

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==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{A}</math> <math>0</math>.
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Draw segment AB with length 10:
 
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<math>A-------10-------B</math>
 
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To have area 100, we need the height to be 20, but there is no way to do this when the sum of the other sides is <math>50-10=40</math>.
 
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--mguempel
 
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==Solution 3==
 
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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. If we try to choose any point <math>C</math> by the Triangle Inequality it will be greater than 10 + 20 + 20, so the answer is <math>\boxed{A}</math> <math>0</math>.
 
  
 
==See Also==
 
==See Also==

Revision as of 18:40, 14 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. 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 $\sqrt{20^2+5^2} \Rightarrow \sqrt{425}$. The perimeter of this minimized triangle is $2\sqrt{425} + 10$, which is larger than 50. Since the minimized perimeter is greater than 50, there is no triangle that satisfies the condition, giving us $\boxed{A) 0}$

iron

Solution 2

WLOG let $AB$ be a horizontal segment of length 10. Now, realize $C$ 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, WLOG $AC<20$. Dropping altitude $CD$ we have a right triangle $ACD$ with hypotenuse $AC<20$ and leg $CD=20$ which is clearly impossible, giving us $\boxed{A}$ $0$.

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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