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Difference between revisions of "2000 AMC 8 Problems/Problem 25"

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The area of rectangle <math>ABCD</math> is <math>72</math>. If point <math>A</math> and the midpoints of <math> \overline{BC} </math> and <math> \overline{CD} </math> are joined to form a triangle, the area of that triangle is
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==Problem==
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The area of rectangle <math>ABCD</math> is <math>72</math> units squared. If point <math>A</math> and the midpoints of <math> \overline{BC} </math> and <math> \overline{CD} </math> are joined to form a triangle, the area of that triangle is
  
 
<asy>
 
<asy>
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<math> \text{(A)}\ 21\qquad\text{(B)}\ 27\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 </math>
 
<math> \text{(A)}\ 21\qquad\text{(B)}\ 27\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 </math>
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==Solution 1==
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To quickly solve this multiple choice problem, make the (not necessarily valid, but very convenient) assumption that <math>ABCD</math> can have any dimension.  Give the rectangle dimensions of <math>AB = CD = 12</math> and <math>BC = AD= 6</math>, which is the easiest way to avoid fractions.  Labelling the right midpoint as <math>M</math>, and the bottom midpoint as <math>N</math>, we know that <math>DN = NC = 6</math>, and <math>BM = MC = 3</math>. 
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<math>[\triangle ADN] = \frac{1}{2}\cdot 6\cdot 6 = 18</math>
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<math>[\triangle MNC] = \frac{1}{2}\cdot 3\cdot 6 = 9</math>
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<math>[\triangle ABM] = \frac{1}{2}\cdot 12\cdot 3 = 18</math>
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<math>[\triangle AMN] = [\square ABCD] - [\triangle ADN] - [\triangle MNC] - [\triangle ABM]</math>
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<math>[\triangle AMN] = 72 - 18 - 9 - 18</math>
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<math>[\triangle AMN] = 27</math>, and the answer is <math>\boxed{B}</math>
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==Solution 2==
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The above answer is fast, but satisfying, and assumes that the area of <math>\triangle AMN</math> is independent of the dimensions of the rectangle. All in all, it's a very good answer though. However this is an alternative if you don't get the above answer. Label <math>AB = CD = l</math> and <math>BC = DA = h</math>
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Labelling <math>m</math> and <math>n</math> as the right and lower midpoints respectively, and redoing all the work above, we get:
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<math>[\triangle ADN] = \frac{1}{2}\cdot h\cdot \frac{l}{2} = \frac{lh}{4}</math>
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<math>[\triangle MNC] = \frac{1}{2}\cdot \frac{l}{2}\cdot \frac{h}{2} = \frac{lh}{8}</math>
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<math>[\triangle ABM] = \frac{1}{2}\cdot l\cdot \frac{h}{2} = \frac{lh}{4}</math>
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<math>[\triangle AMN] = [\square ABCD] - [\triangle ADN] - [\triangle MNC] - [\triangle ABM]</math>
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<math>[\triangle AMN] = lh - \frac{lh}{4} - \frac{lh}{8} - \frac{lh}{4}</math>
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<math>[\triangle AMN] = \frac{3}{8}lh = \frac{3}{8}\cdot 72 = 27</math>, and the answer is <math>\boxed{B}</math>
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==Solution 3==
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Let's assume WLOG that the sides of the rectangle are <math>9</math> and <math>8.</math> The area of the 3 triangles would then be <math>8\cdot\frac{9}{2}\cdot\frac{1}{2} = 18,</math>
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<math>4\cdot\frac{9}{2}\cdot\frac{1}{2} = 9,</math>
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<math>4\cdot 9\cdot\frac{1}{2} = 18.</math>
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Adding these up, we get <math>45</math>, and subtracting that from <math>72</math>, we get <math>27</math>, so the answer is <math>\boxed{B}</math>
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~ilee0820
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== Video Solution ==
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https://youtu.be/yoIO9q_GTig. Soo, DRMS, NM
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https://www.youtube.com/watch?v=XxQwfirFn4M    ~David
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{{AMC8 box|year=2000|num-b=24|after=Last<br>Question}}
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{{MAA Notice}}

Latest revision as of 13:48, 26 December 2024

Problem

The area of rectangle $ABCD$ is $72$ units squared. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is

[asy] pair A,B,C,D; A = (0,8); B = (9,8); C = (9,0); D = (0,0); draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW);[/asy]

$\text{(A)}\ 21\qquad\text{(B)}\ 27\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40$

Solution 1

To quickly solve this multiple choice problem, make the (not necessarily valid, but very convenient) assumption that $ABCD$ can have any dimension. Give the rectangle dimensions of $AB = CD = 12$ and $BC = AD= 6$, which is the easiest way to avoid fractions. Labelling the right midpoint as $M$, and the bottom midpoint as $N$, we know that $DN = NC = 6$, and $BM = MC = 3$.

$[\triangle ADN] = \frac{1}{2}\cdot 6\cdot 6 = 18$

$[\triangle MNC] = \frac{1}{2}\cdot 3\cdot 6 = 9$

$[\triangle ABM] = \frac{1}{2}\cdot 12\cdot 3 = 18$

$[\triangle AMN] = [\square ABCD] - [\triangle ADN] - [\triangle MNC] - [\triangle ABM]$

$[\triangle AMN] = 72 - 18 - 9 - 18$

$[\triangle AMN] = 27$, and the answer is $\boxed{B}$

Solution 2

The above answer is fast, but satisfying, and assumes that the area of $\triangle AMN$ is independent of the dimensions of the rectangle. All in all, it's a very good answer though. However this is an alternative if you don't get the above answer. Label $AB = CD = l$ and $BC = DA = h$

Labelling $m$ and $n$ as the right and lower midpoints respectively, and redoing all the work above, we get:

$[\triangle ADN] = \frac{1}{2}\cdot h\cdot \frac{l}{2} = \frac{lh}{4}$

$[\triangle MNC] = \frac{1}{2}\cdot \frac{l}{2}\cdot \frac{h}{2} = \frac{lh}{8}$

$[\triangle ABM] = \frac{1}{2}\cdot l\cdot \frac{h}{2} = \frac{lh}{4}$

$[\triangle AMN] = [\square ABCD] - [\triangle ADN] - [\triangle MNC] - [\triangle ABM]$

$[\triangle AMN] = lh - \frac{lh}{4} - \frac{lh}{8} - \frac{lh}{4}$

$[\triangle AMN] = \frac{3}{8}lh = \frac{3}{8}\cdot 72 = 27$, and the answer is $\boxed{B}$

Solution 3

Let's assume WLOG that the sides of the rectangle are $9$ and $8.$ The area of the 3 triangles would then be $8\cdot\frac{9}{2}\cdot\frac{1}{2} = 18,$ $4\cdot\frac{9}{2}\cdot\frac{1}{2} = 9,$ $4\cdot 9\cdot\frac{1}{2} = 18.$ Adding these up, we get $45$, and subtracting that from $72$, we get $27$, so the answer is $\boxed{B}$

~ilee0820

Video Solution

https://youtu.be/yoIO9q_GTig. Soo, DRMS, NM

https://www.youtube.com/watch?v=XxQwfirFn4M ~David


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