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

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{{duplicate|[[2019 AMC 10B Problems|2019 AMC 10B #10]] and [[2019 AMC 12B Problems|2019 AMC 12B #6]]}}
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==Problem==
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In a given plane, points <math>A</math> and <math>B</math> are <math>10</math> units apart. How many points <math>C</math> are there in the plane such that the perimeter of <math>\triangle ABC</math> is <math>50</math> units and the area of <math>\triangle ABC</math> is <math>100</math> square units?
 
In a given plane, points <math>A</math> and <math>B</math> are <math>10</math> units apart. How many points <math>C</math> are there in the plane such that the perimeter of <math>\triangle ABC</math> is <math>50</math> units and the area of <math>\triangle ABC</math> is <math>100</math> square units?
  
 
<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>
  
==Solution==
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==Solution 1==
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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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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
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==Solution 2==
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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>.
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==Solution 3==
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Area = <math>100</math>, perimeter = <math>50</math>, semiperimeter <math>s = 50/2 = 25</math>, <math>z = AB = 10</math>, <math>x = AC</math> and <math>y = 50-10-x = 40-x</math>.
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Using the generic formula for triangle area using semiperimeter <math>s</math> and sides <math>x</math>, <math>y</math>, and <math>z</math>, area = <math>\sqrt{(s)(s-x)(s-y)(s-z)}</math>. (Heron's formula)
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<math>100 = \sqrt{(25)(25-10)(25-x)(25-(40-x))} = \sqrt{(375)(25-x)(x-15)}</math>.
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Square both sides, divide by <math>375</math> and expand the polynomial to get <math>40x - x^2 - 375 = 80/3</math>.
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<math>x^2 -40x + (375+80/3) = 0</math> and the discriminant is <math>((-40)^2 - 4*1*401.\overline{6}) < 0</math> so no real solutions.
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==Video Solution==
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https://youtu.be/7xf_g3YQk00
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~IceMatrix
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https://youtu.be/INvRdwQzC-w
  
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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~savannahsolver
  
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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==See Also==
  
iron
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{{AMC10 box|year=2019|ab=B|num-b=9|num-a=11}}
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{{AMC12 box|year=2019|ab=B|num-b=5|num-a=7}}
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{{MAA Notice}}

Revision as of 17:03, 28 July 2020

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 1

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 $\boxed{\textbf{(A) }0}$.

~IronicNinja

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 $\boxed{\textbf{(A) }0}$.

Solution 3

Area = $100$, perimeter = $50$, semiperimeter $s = 50/2 = 25$, $z = AB = 10$, $x = AC$ and $y = 50-10-x = 40-x$.


Using the generic formula for triangle area using semiperimeter $s$ and sides $x$, $y$, and $z$, area = $\sqrt{(s)(s-x)(s-y)(s-z)}$. (Heron's formula)

$100 = \sqrt{(25)(25-10)(25-x)(25-(40-x))} = \sqrt{(375)(25-x)(x-15)}$.


Square both sides, divide by $375$ and expand the polynomial to get $40x - x^2 - 375 = 80/3$.


$x^2 -40x + (375+80/3) = 0$ and the discriminant is $((-40)^2 - 4*1*401.\overline{6}) < 0$ so no real solutions.

Video Solution

https://youtu.be/7xf_g3YQk00

~IceMatrix

https://youtu.be/INvRdwQzC-w

~savannahsolver

See Also

2019 AMC 10B (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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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