Difference between revisions of "2013 AMC 10A Problems/Problem 18"

(Solution)
(Solution)
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==Solution==
 
==Solution==
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<center><asy>
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size(8cm);
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pair A, B, C, D, E, EE;
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A = (0,0);
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B = (1,2);
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C = (3,3);
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D = (4,0);
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E = (27/8,15/8);
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EE = (27/8,0);
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draw(A--B--C--D--A--E);
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draw(E--EE,linetype("8 8"));
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dot(A);
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dot(B);
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dot(C);
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dot(D);
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dot(E);
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draw(rightanglemark(E,EE,D,4));
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label("A",A,SW);
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label("B",B,NW);
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label("C",C,NE);
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label("D",D,SE);
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label("E",E,NE);
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label("$4$",(A+D)/2,S);
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label("$\frac{27}{8}$",(D+EE)/2,S);
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label("$\frac{15}{8}$",(E+EE)/2,W);
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</asy></center>
  
 
First, we shall find the area of quadrilateral <math>ABCD</math>. This can be done in any of three ways:
 
First, we shall find the area of quadrilateral <math>ABCD</math>. This can be done in any of three ways:
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Line CD can be expressed as <math>y = -3x+12</math>, so the <math>x</math> coordinate of E satisfies <math>\frac{15}{8} = -3x + 12</math>.  Solving for <math>x</math>, we find that <math>x = \frac{27}{8}</math>.
 
Line CD can be expressed as <math>y = -3x+12</math>, so the <math>x</math> coordinate of E satisfies <math>\frac{15}{8} = -3x + 12</math>.  Solving for <math>x</math>, we find that <math>x = \frac{27}{8}</math>.
  
From this, we know that <math>E = (\frac{27}{8}, \frac{15}{8})</math>.  <math>27 + 15 + 8 + 8 = \boxed{\textbf{(B) }58}</math>
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From this, we know that <math>E = \left(\frac{27}{8}, \frac{15}{8}\right)</math>.  <math>27 + 15 + 8 + 8 = \boxed{\textbf{(B) }58}</math>
  
 
==See Also==
 
==See Also==

Revision as of 17:09, 27 December 2017

Problem

Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $(\frac{p}{q}, \frac{r}{s})$, where these fractions are in lowest terms. What is $p+q+r+s$?


$\textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75$

Solution

[asy] size(8cm); pair A, B, C, D, E, EE; A = (0,0); B = (1,2); C = (3,3); D = (4,0); E = (27/8,15/8); EE = (27/8,0); draw(A--B--C--D--A--E); draw(E--EE,linetype("8 8")); dot(A); dot(B); dot(C); dot(D); dot(E); draw(rightanglemark(E,EE,D,4)); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,NE); label("$4$",(A+D)/2,S); label("$\frac{27}{8}$",(D+EE)/2,S); label("$\frac{15}{8}$",(E+EE)/2,W); [/asy]

First, we shall find the area of quadrilateral $ABCD$. This can be done in any of three ways:

Pick's Theorem: $[ABCD] = I + \dfrac{B}{2} - 1 = 5 + \dfrac{7}{2} - 1 = \dfrac{15}{2}.$

Splitting: Drop perpendiculars from $B$ and $C$ to the x-axis to divide the quadrilateral into triangles and trapezoids, and so the area is $1 + 5 + \dfrac{3}{2} = \dfrac{15}{2}.$

Shoelace Method: The area is half of $|1 \cdot 3 - 2 \cdot 3 - 3 \cdot 4| = 15$, or $\dfrac{15}{2}$.

$[ABCD] = \frac{15}{2}$. Therefore, each equal piece that the line separates $ABCD$ into must have an area of $\frac{15}{4}$.

Call the point where the line through $A$ intersects $\overline{CD}$ $E$. We know that $[ADE] = \frac{15}{4} = \frac{bh}{2}$. Furthermore, we know that $b = 4$, as $AD = 4$. Thus, solving for $h$, we find that $2h = \frac{15}{4}$, so $h = \frac{15}{8}$. This gives that the y coordinate of E is $\frac{15}{8}$.

Line CD can be expressed as $y = -3x+12$, so the $x$ coordinate of E satisfies $\frac{15}{8} = -3x + 12$. Solving for $x$, we find that $x = \frac{27}{8}$.

From this, we know that $E = \left(\frac{27}{8}, \frac{15}{8}\right)$. $27 + 15 + 8 + 8 = \boxed{\textbf{(B) }58}$

See Also

2013 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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
2013 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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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