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Difference between revisions of "2004 AMC 12A Problems/Problem 8"

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{{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #8]] and [[2004 AMC 10A Problems/Problem 9|2004 AMC 10A #9]]}}
 
{{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #8]] and [[2004 AMC 10A Problems/Problem 9|2004 AMC 10A #9]]}}
 +
 
== Problem ==
 
== Problem ==
 
In the overlapping [[triangle]]s <math>\triangle{ABC}</math> and <math>\triangle{ABE}</math> sharing common [[edge | side]] <math>AB</math>, <math>\angle{EAB}</math> and <math>\angle{ABC}</math> are [[right angle]]s, <math>AB=4</math>, <math>BC=6</math>, <math>AE=8</math>, and <math>\overline{AC}</math> and <math>\overline{BE}</math> intersect at <math>D</math>. What is the difference between the areas of <math>\triangle{ADE}</math> and <math>\triangle{BDC}</math>?  
 
In the overlapping [[triangle]]s <math>\triangle{ABC}</math> and <math>\triangle{ABE}</math> sharing common [[edge | side]] <math>AB</math>, <math>\angle{EAB}</math> and <math>\angle{ABC}</math> are [[right angle]]s, <math>AB=4</math>, <math>BC=6</math>, <math>AE=8</math>, and <math>\overline{AC}</math> and <math>\overline{BE}</math> intersect at <math>D</math>. What is the difference between the areas of <math>\triangle{ADE}</math> and <math>\triangle{BDC}</math>?  
  
−
[[Image:AMC10_2004A_9.gif|center]]
+
<asy>
 +
size(150);
 +
defaultpen(linewidth(0.4));
 +
//Variable Declarations
 +
pair A, B, C, D, E;
 +
 
 +
//Variable Definitions
 +
A=(0, 0);
 +
B=(4, 0);
 +
C=(4, 6);
 +
E=(0, 8);
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D=extension(A,C,B,E);
 +
 
 +
//Initial Diagram
 +
draw(A--B--C--A--E--B);
 +
label("$A$",A,SW);
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label("$B$",B,SE);
 +
label("$C$",C,NE);
 +
label("$D$",D,3N);
 +
label("$E$",E,NW);
 +
 
 +
//Side labels
 +
label("$4$",A--B,S);
 +
label("$8$",A--E,W);
 +
label("$6$",B--C,ENE);
 +
</asy>
  
 
<math>\mathrm {(A)}\ 2 \qquad \mathrm {(B)}\ 4 \qquad \mathrm {(C)}\ 5 \qquad \mathrm {(D)}\ 8 \qquad \mathrm {(E)}\ 9 \qquad</math>
 
<math>\mathrm {(A)}\ 2 \qquad \mathrm {(B)}\ 4 \qquad \mathrm {(C)}\ 5 \qquad \mathrm {(D)}\ 8 \qquad \mathrm {(E)}\ 9 \qquad</math>
  
−
__TOC__
+
== Solutions ==
−
== Solution 1 ==
+
===Solution 1===
 +
Looking, we see that the area of <math>[\triangle EBA]</math> is 16 and the area of <math>[\triangle ABC]</math> is 12. Set the area of <math>[\triangle ADB]</math> to be x. We want to find <math>[\triangle ADE]</math> - <math>[\triangle CDB]</math>. So, that would be <math>[\triangle EBA]-[\triangle ADB]=16-x</math> and <math>[\triangle ABC]-[\triangle ADB]=12-x</math>. Therefore, <math>[\triangle ADE]-[\triangle DBC]=(16-x)-(12-x)=16-x-12+x= \boxed{\mathrm{(B)}\ 4}</math>
  
 +
~ MathKatana
 +
 +
=== Solution 2 ===
 
Since <math>AE \perp AB</math> and <math>BC \perp AB</math>, <math>AE \parallel BC</math>. By alternate interior angles and <math>AA\sim</math>, we find that <math>\triangle ADE \sim \triangle CDB</math>, with side length ratio <math>\frac{4}{3}</math>. Their heights also have the same ratio, and since the two heights add up to <math>4</math>, we have that <math>h_{ADE} = 4 \cdot \frac{4}{7} = \frac{16}{7}</math> and <math>h_{CDB} = 3 \cdot \frac 47 = \frac {12}7</math>. Subtracting the areas, <math>\frac{1}{2} \cdot 8 \cdot \frac {16}7 - \frac 12 \cdot 6 \cdot \frac{12}7 = 4</math> <math>\Rightarrow</math> <math>\boxed{\mathrm{(B)}\ 4}</math>.
 
Since <math>AE \perp AB</math> and <math>BC \perp AB</math>, <math>AE \parallel BC</math>. By alternate interior angles and <math>AA\sim</math>, we find that <math>\triangle ADE \sim \triangle CDB</math>, with side length ratio <math>\frac{4}{3}</math>. Their heights also have the same ratio, and since the two heights add up to <math>4</math>, we have that <math>h_{ADE} = 4 \cdot \frac{4}{7} = \frac{16}{7}</math> and <math>h_{CDB} = 3 \cdot \frac 47 = \frac {12}7</math>. Subtracting the areas, <math>\frac{1}{2} \cdot 8 \cdot \frac {16}7 - \frac 12 \cdot 6 \cdot \frac{12}7 = 4</math> <math>\Rightarrow</math> <math>\boxed{\mathrm{(B)}\ 4}</math>.
  
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=== Solution 2 ===
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=== Solution 3 ===
 
Let <math>[X]</math> represent the area of figure <math>X</math>. Note that <math>[\triangle BEA]=[\triangle ABD]+[\triangle ADE]</math> and <math>[\triangle BCA]=[\triangle ABD]+[\triangle BDC]</math>.
 
Let <math>[X]</math> represent the area of figure <math>X</math>. Note that <math>[\triangle BEA]=[\triangle ABD]+[\triangle ADE]</math> and <math>[\triangle BCA]=[\triangle ABD]+[\triangle BDC]</math>.
  
−
<math>[\triangle ADE]-[\triangle BDC]=[\triangle BEA]-[\triangle BCA]=\frac{1}{2} 8*4-\frac{1}{2} 6*4= 16-12=4</math> <math>\Rightarrow</math> <math>\boxed{(B)}</math>.
+
<math>[\triangle ADE]-[\triangle BDC]=[\triangle BEA]-[\triangle BCA]=\frac{1}{2}\times8\times4-\frac{1}{2}\times6\times4= 16-12=4\Rightarrow\boxed{\mathrm{(B)}\ 4}</math>
 +
 
 +
=== Solution 4 (coordbash)===
 +
Put figure <math>ABCDE</math> on a graph. <math>\overline{AC}</math> goes from (0, 0) to (4, 6) and <math>\overline{BE}</math> goes from (4, 0) to (0, 8). <math>\overline{AC}</math> is on line <math>y = 1.5x</math>. <math>\overline{BE}</math> is on line <math>y = -2x + 8</math>. Finding intersection between these points,
 +
 
 +
<math>1.5x = -2x + 8</math>.
 +
 
 +
<math>3.5x = 8 </math>
 +
 
 +
<math> x = 8 \times \frac{2}{7}</math>
 +
 
 +
<math> = \frac{16}{7}</math>
 +
 
 +
This gives us the x-coordinate of D.
 +
So, <math>\frac{16}{7}</math> is the height of <math>\triangle ADE</math>, then area of <math>\triangle ADE</math> is
 +
<math>\frac{16}{7} \times 8 \times \frac{1}{2}</math>
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<math> = \frac{64}{7}</math>
 +
 
 +
Now, the height of <math>\triangle BDC</math> is <math>4-\frac{16}{7} = \frac{12}{7}</math>
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And the area of <math>\triangle BDC</math> is <math>6 \times \frac{12}{7} \times \frac{1}{2} = \frac{36}{7}</math>
 +
 
 +
This gives us <math>\frac{64}{7} - \frac{36}{7} = 4</math>
 +
 
 +
Therefore, the difference is <math>4</math>
 +
 
 +
=== Solution 5 ===
 +
We want to figure out <math>[\triangle ADE] - [\triangle BDC]</math>.
 +
Notice that <math>\triangle ABC</math> and <math>\triangle BAE</math> "intersect" and form <math>\triangle ADB</math>.
 +
 
 +
This means that <math>[\triangle BAE] - [\triangle ABC)] = [\triangle ADE] - [\triangle BDC]</math> because <math>[\triangle ADB]</math> cancels out, which can be seen easily in the diagram.
 +
 
 +
<math>[\triangle BAE] = 0.5 \cdot 4 \cdot 8 = 16</math>
 +
 
 +
<math>[\triangle ABC] = 0.5 \cdot 4 \cdot 16 = 12</math>
 +
 
 +
<math>[\triangle BDC] - [\triangle ADE] = 16 - 12 =\boxed{\mathrm{(B)}\ 4}</math>
 +
 
 +
==Video Solution==
 +
https://youtu.be/DlA71MBSviU
 +
 
 +
Education, the Study of Everything
  
  

Latest revision as of 22:24, 28 September 2024

The following problem is from both the 2004 AMC 12A #8 and 2004 AMC 10A #9, so both problems redirect to this page.

Problem

In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$, $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB=4$, $BC=6$, $AE=8$, and $\overline{AC}$ and $\overline{BE}$ intersect at $D$. What is the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$?

[asy] size(150); defaultpen(linewidth(0.4)); //Variable Declarations pair A, B, C, D, E; //Variable Definitions A=(0, 0); B=(4, 0); C=(4, 6); E=(0, 8); D=extension(A,C,B,E); //Initial Diagram draw(A--B--C--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,3N); label("$E$",E,NW); //Side labels label("$4$",A--B,S); label("$8$",A--E,W); label("$6$",B--C,ENE); [/asy]

$\mathrm {(A)}\ 2 \qquad \mathrm {(B)}\ 4 \qquad \mathrm {(C)}\ 5 \qquad \mathrm {(D)}\ 8 \qquad \mathrm {(E)}\ 9 \qquad$

Solutions

Solution 1

Looking, we see that the area of $[\triangle EBA]$ is 16 and the area of $[\triangle ABC]$ is 12. Set the area of $[\triangle ADB]$ to be x. We want to find $[\triangle ADE]$ - $[\triangle CDB]$. So, that would be $[\triangle EBA]-[\triangle ADB]=16-x$ and $[\triangle ABC]-[\triangle ADB]=12-x$. Therefore, $[\triangle ADE]-[\triangle DBC]=(16-x)-(12-x)=16-x-12+x= \boxed{\mathrm{(B)}\ 4}$

~ MathKatana

Solution 2

Since $AE \perp AB$ and $BC \perp AB$, $AE \parallel BC$. By alternate interior angles and $AA\sim$, we find that $\triangle ADE \sim \triangle CDB$, with side length ratio $\frac{4}{3}$. Their heights also have the same ratio, and since the two heights add up to $4$, we have that $h_{ADE} = 4 \cdot \frac{4}{7} = \frac{16}{7}$ and $h_{CDB} = 3 \cdot \frac 47 = \frac {12}7$. Subtracting the areas, $\frac{1}{2} \cdot 8 \cdot \frac {16}7 - \frac 12 \cdot 6 \cdot \frac{12}7 = 4$ $\Rightarrow$ $\boxed{\mathrm{(B)}\ 4}$.

Solution 3

Let $[X]$ represent the area of figure $X$. Note that $[\triangle BEA]=[\triangle ABD]+[\triangle ADE]$ and $[\triangle BCA]=[\triangle ABD]+[\triangle BDC]$.

$[\triangle ADE]-[\triangle BDC]=[\triangle BEA]-[\triangle BCA]=\frac{1}{2}\times8\times4-\frac{1}{2}\times6\times4= 16-12=4\Rightarrow\boxed{\mathrm{(B)}\ 4}$

Solution 4 (coordbash)

Put figure $ABCDE$ on a graph. $\overline{AC}$ goes from (0, 0) to (4, 6) and $\overline{BE}$ goes from (4, 0) to (0, 8). $\overline{AC}$ is on line $y = 1.5x$. $\overline{BE}$ is on line $y = -2x + 8$. Finding intersection between these points,

$1.5x = -2x + 8$.

$3.5x = 8$

$x = 8 \times \frac{2}{7}$

$= \frac{16}{7}$

This gives us the x-coordinate of D. So, $\frac{16}{7}$ is the height of $\triangle ADE$, then area of $\triangle ADE$ is $\frac{16}{7} \times 8 \times \frac{1}{2}$ $= \frac{64}{7}$

Now, the height of $\triangle BDC$ is $4-\frac{16}{7} = \frac{12}{7}$ And the area of $\triangle BDC$ is $6 \times \frac{12}{7} \times \frac{1}{2} = \frac{36}{7}$

This gives us $\frac{64}{7} - \frac{36}{7} = 4$

Therefore, the difference is $4$

Solution 5

We want to figure out $[\triangle ADE] - [\triangle BDC]$. Notice that $\triangle ABC$ and $\triangle BAE$ "intersect" and form $\triangle ADB$.

This means that $[\triangle BAE] - [\triangle ABC)] = [\triangle ADE] - [\triangle BDC]$ because $[\triangle ADB]$ cancels out, which can be seen easily in the diagram.

$[\triangle BAE] = 0.5 \cdot 4 \cdot 8 = 16$

$[\triangle ABC] = 0.5 \cdot 4 \cdot 16 = 12$

$[\triangle BDC] - [\triangle ADE] = 16 - 12 =\boxed{\mathrm{(B)}\ 4}$

Video Solution

https://youtu.be/DlA71MBSviU

Education, the Study of Everything


See also

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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
2004 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

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