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

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== Solutions ==
 
== Solutions ==
 
===Solution 0===
 
===Solution 0===
−
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></math>[\triangle ADE]-[\triangle DBC]=(16-x)-(12-x)=16-x-12+x= \boxed{\mathrm{(B)}\ 4}<math>
+
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
 
~ MathKatana
  
 
=== Solution 1 ===
 
=== Solution 1 ===
−
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>.
  
 
=== Solution 2 ===
 
=== Solution 2 ===
−
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}\times8\times4-\frac{1}{2}\times6\times4= 16-12=4\Rightarrow\boxed{\mathrm{(B)}\ 4}<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 3 (coordbash)===
 
=== Solution 3 (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,
+
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>1.5x = -2x + 8</math>.  
  
−
</math>3.5x = 8 <math>
+
<math>3.5x = 8 </math>
  
−
</math> x = 8 \times \frac{2}{7}<math>
+
<math> x = 8 \times \frac{2}{7}</math>
  
−
</math> = \frac{16}{7}<math>
+
<math> = \frac{16}{7}</math>
  
 
This gives us the x-coordinate of D.  
 
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  
+
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>
+
<math>\frac{16}{7} \times 8 \times \frac{1}{2}</math>
−
</math> = \frac{64}{7}<math>
+
<math> = \frac{64}{7}</math>
  
−
Now, the height of </math>\triangle BDC<math> is </math>4-\frac{16}{7} = \frac{12}{7}<math>
+
Now, the height of <math>\triangle BDC</math> is <math>4-\frac{16}{7} = \frac{12}{7}</math>
−
And the area of </math>\triangle BDC<math> is </math>6 \times \frac{12}{7} \times \frac{1}{2} = \frac{36}{7}<math>
+
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>
+
This gives us <math>\frac{64}{7} - \frac{36}{7} = 4</math>
  
−
Therefore, the difference is </math>4<math>
+
Therefore, the difference is <math>4</math>
  
 
=== Solution 4 ===
 
=== Solution 4 ===
−
We want to figure out </math>Area(\triangle ADE) - Area(\triangle BDC)<math>.
+
We want to figure out <math>Area(\triangle ADE) - Area(\triangle BDC)</math>.
−
Notice that </math>\triangle ABC<math> and </math>\triangle BAE<math> "intersect" and form </math>\triangle ADB<math>.
+
Notice that <math>\triangle ABC</math> and <math>\triangle BAE</math> "intersect" and form <math>\triangle ADB</math>.
  
−
This means that </math>Area(\triangle BAE) - Area(\triangle ABC) = Area(\triangle ADE) - Area(\triangle BDC)<math> because </math>Area(\triangle ADB)<math> cancels out, which can be seen easily in the diagram.
+
This means that <math>Area(\triangle BAE) - Area(\triangle ABC) = Area(\triangle ADE) - Area(\triangle BDC)</math> because <math>Area(\triangle ADB)</math> cancels out, which can be seen easily in the diagram.
  
−
</math>Area(\triangle BAE) = 0.5 * 4 * 8 = 16<math>
+
<math>Area(\triangle BAE) = 0.5 * 4 * 8 = 16</math>
  
−
</math>Area(\triangle ABC) = 0.5 * 4 * 16 = 12<math>
+
<math>Area(\triangle ABC) = 0.5 * 4 * 16 = 12</math>
  
−
</math>Area(\triangle BDC) - Area(\triangle ADE) = 16 - 12 =\boxed{\mathrm{(B)}\ 4}$
+
<math>Area(\triangle BDC) - Area(\triangle ADE) = 16 - 12 =\boxed{\mathrm{(B)}\ 4}</math>
  
 
==Video Solution==
 
==Video Solution==

Revision as of 20:15, 5 April 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 0

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 1

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 2

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 3 (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 4

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

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

$Area(\triangle BAE) = 0.5 * 4 * 8 = 16$

$Area(\triangle ABC) = 0.5 * 4 * 16 = 12$

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

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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