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Difference between revisions of "1982 AHSME Problems/Problem 9"

(Created page with "== Solution for Problem 9 == <asy> size(350); defaultpen(fontsize(10)); pair A=origin, O=(10,0), B=(3,0), N=(0,5), C=(3,5), P=(5,0), D=(1,1), G=(9,1), F=(1,0); draw(G--A--D--...")
 
(Add problem statement)
 
(One intermediate revision by one other user not shown)
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== Solution for Problem 9 ==
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== Problem ==
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A vertical line divides the triangle with vertices <math>(0,0), (1,1)</math>, and <math>(9,1)</math> in the <math>xy\text{-plane}</math> into two regions of equal area.
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The equation of the line is <math>x=</math>
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<math>\textbf {(A)}\ 2.5 \qquad
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\textbf {(B)}\ 3.0 \qquad
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\textbf {(C)}\ 3.5 \qquad
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\textbf {(D)}\ 4.0\qquad
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\textbf {(E)}\ 4.5  </math> 
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== Solution ==
  
 
<asy>
 
<asy>
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label("$(1,1)$", D, NE);</asy>
 
label("$(1,1)$", D, NE);</asy>
 
The vertical line that divides <math>\triangle ABC</math> into two equal regions has equation <math>x=a,</math> as shown in the diagram.
 
The vertical line that divides <math>\triangle ABC</math> into two equal regions has equation <math>x=a,</math> as shown in the diagram.
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The area of <math>ABC</math> is half of the height times <math>AC,</math> so because the Y coordinate of A is 1, and <math>\overline{AF}</math> is the height, because the difference of the x coordinates between <math>A</math> and <math>C</math> is <math>8,</math> we have <math>[ABC]=1/2 \cdot 8 \cdot 1 = 4.</math> Thus the two regions must have area <math>2</math> each.
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The area of <math>ABC</math> is half of the height times <math>AC,</math> so because the Y coordinate of A is 1 and <math>\overline{AF}</math> is the height, because the difference of the x coordinates between <math>A</math> and <math>C</math> is <math>8,</math> we have <math>[ABC]=1/2 \cdot 8 \cdot 1 = 4.</math> Thus the two regions must have area <math>2</math> each.
  
 
Since <math>\triangle ABF</math> has area <math>1/2,</math> we know that the portion of <math>\triangle ABC</math> made by the points <math>A,</math> <math>B</math> and the intersection <math>\overline{AF}</math> and <math>\overline{BC}</math> will be less than <math>1/2,</math> which is less than half of the triangle's area, or 2. Therefore <math>\overline{AF}</math> is to the left of vertical line <math>x=a</math> (Passing through point <math>E</math>).
 
Since <math>\triangle ABF</math> has area <math>1/2,</math> we know that the portion of <math>\triangle ABC</math> made by the points <math>A,</math> <math>B</math> and the intersection <math>\overline{AF}</math> and <math>\overline{BC}</math> will be less than <math>1/2,</math> which is less than half of the triangle's area, or 2. Therefore <math>\overline{AF}</math> is to the left of vertical line <math>x=a</math> (Passing through point <math>E</math>).
  
 
The equation of line BC is <math>y=x/9,</math> and the vertical line <math>x=a</math> intersects <math>\overline{BC}</math> at the point <math>(a, a/9).</math> Because the area of the portion of <math>\triangle ABC</math> on the right is 2, we have <cmath>2=1/2(1-a/9)(9-a)</cmath> or <cmath>(9-a)^2=36.</cmath> Therefore <math>a>0</math> so <math>a=3=x.</math>
 
The equation of line BC is <math>y=x/9,</math> and the vertical line <math>x=a</math> intersects <math>\overline{BC}</math> at the point <math>(a, a/9).</math> Because the area of the portion of <math>\triangle ABC</math> on the right is 2, we have <cmath>2=1/2(1-a/9)(9-a)</cmath> or <cmath>(9-a)^2=36.</cmath> Therefore <math>a>0</math> so <math>a=3=x.</math>

Latest revision as of 12:30, 16 July 2024

Problem

A vertical line divides the triangle with vertices $(0,0), (1,1)$, and $(9,1)$ in the $xy\text{-plane}$ into two regions of equal area. The equation of the line is $x=$

$\textbf {(A)}\ 2.5 \qquad \textbf {(B)}\ 3.0 \qquad \textbf {(C)}\ 3.5 \qquad \textbf {(D)}\ 4.0\qquad \textbf {(E)}\ 4.5$

Solution

[asy] size(350); defaultpen(fontsize(10)); pair A=origin, O=(10,0), B=(3,0), N=(0,5), C=(3,5), P=(5,0), D=(1,1), G=(9,1), F=(1,0); draw(G--A--D--cycle, linewidth(0.7)); draw(D--F, linewidth(0.6)); draw(B--C, linewidth(0.5)); draw(A--N, linewidth(0.4)); draw(A--O, linewidth(0.4)); dot(A^^B^^D^^G^^F); label("$A$", D, W); label("$B$", A, dir(0)); label("$C$", G, dir(100)); label("$E$", B, SE); label("$F$", F, SE); label("$(9,1)$", G, SE); label("$(0,0)$", A, SW); label("$8$", D--G, dir(40)); label("$y=x/9$", A--G, SE); label("$x=a$", B--N, SE); label("$(1,1)$", D, NE);[/asy] The vertical line that divides $\triangle ABC$ into two equal regions has equation $x=a,$ as shown in the diagram. The area of $ABC$ is half of the height times $AC,$ so because the Y coordinate of A is 1 and $\overline{AF}$ is the height, because the difference of the x coordinates between $A$ and $C$ is $8,$ we have $[ABC]=1/2 \cdot 8 \cdot 1 = 4.$ Thus the two regions must have area $2$ each.

Since $\triangle ABF$ has area $1/2,$ we know that the portion of $\triangle ABC$ made by the points $A,$ $B$ and the intersection $\overline{AF}$ and $\overline{BC}$ will be less than $1/2,$ which is less than half of the triangle's area, or 2. Therefore $\overline{AF}$ is to the left of vertical line $x=a$ (Passing through point $E$).

The equation of line BC is $y=x/9,$ and the vertical line $x=a$ intersects $\overline{BC}$ at the point $(a, a/9).$ Because the area of the portion of $\triangle ABC$ on the right is 2, we have \[2=1/2(1-a/9)(9-a)\] or \[(9-a)^2=36.\] Therefore $a>0$ so $a=3=x.$

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