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Difference between revisions of "2000 AMC 12 Problems/Problem 19"

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== Solution 1==
 
== Solution 1==
 
The answer is exactly <math>3</math>, choice <math>\mathrm{(C)}</math>.
 
The answer is exactly <math>3</math>, choice <math>\mathrm{(C)}</math>.
−
We can find the area of triangle <math>ADE</math> by using the simple formula <math>\frac{bh}{2}</math>. Dropping an altitude from <math>A</math>, we see that it has length <math>12</math> ( we can split the large triangle into a <math>9-12-15</math> and a <math>5-12-13</math> triangle). Then we can apply the [[Angle Bisector Theorem]] on triangle <math>ABC</math> to solve for <math>BE</math>. Solving <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, we get that <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. Applying the formula <math>\frac{bh}{2}</math>, we get <math>\frac{12*\frac{1}{2}}{2}=3</math>.
+
We can find the area of triangle <math>ADE</math> by using the simple formula <math>\frac{bh}{2}</math>. Dropping an altitude from <math>A</math>, we see that it has length <math>12</math> (we can split the large triangle into a <math>9-12-15</math> and a <math>5-12-13</math> triangle). Then we can apply the [[Angle Bisector Theorem]] on triangle <math>ABC</math> to solve for <math>BE</math>. Solving <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, we get that <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. Applying the formula <math>\frac{bh}{2}</math>, we get <math>\frac{12*\frac{1}{2}}{2}=3</math>.
  
 
==Solution 2==
 
==Solution 2==
−
The area of <math>ADE</math> is <math>\frac{DE\cdot h}{2}=\frac{DE}{BC} \cdot \frac{BC\cdot h}{2}=\frac{DE}{BC}[ABC]</math> where <math>h</math> is the height of triangle <math>ABC</math>. Using Angle Bisector Theorem, we find <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, which we solve to get <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. We can now use Heron's Formula on <math>ABC</math>.
+
The area of <math>ADE</math> is <math>\frac{DE\cdot h}{2}=\frac{DE}{BC} \cdot \frac{BC\cdot h}{2}=\frac{DE}{BC}[ABC]</math> where <math>h</math> is the height of triangle <math>ABC</math>. Using Angle Bisector Theorem, we find <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, which we solve to get <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. We can now use [[Heron's Formula]] on <math>ABC</math>.
 
<cmath>s=\frac{AB+BC+AC}{2}=21</cmath>
 
<cmath>s=\frac{AB+BC+AC}{2}=21</cmath>
 
<cmath>[ABC]=\sqrt{(s)(s-AB)(s-BC)(s-AC)}=\sqrt{(21)(8)(7)(6)}=84</cmath>
 
<cmath>[ABC]=\sqrt{(s)(s-AB)(s-BC)(s-AC)}=\sqrt{(21)(8)(7)(6)}=84</cmath>
 
<cmath>\frac{DE}{BC}[ABC]=\frac{\frac{1}{2}}{14}\cdot 84=3</cmath>
 
<cmath>\frac{DE}{BC}[ABC]=\frac{\frac{1}{2}}{14}\cdot 84=3</cmath>
 
Therefore, the answer is <math>\mathrm{C}</math>.
 
Therefore, the answer is <math>\mathrm{C}</math>.
−
 
  
 
==Solution 3==
 
==Solution 3==
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pair A,B,C,D,E;
 
pair A,B,C,D,E;
 
B=(0,0);
 
B=(0,0);
−
C=(14;0);
+
C=(14,0);
 
A=intersectionpoint(arc(B,13,0,90),arc(C,15,90,180));
 
A=intersectionpoint(arc(B,13,0,90),arc(C,15,90,180));
 
draw(A--B--C--cycle);
 
draw(A--B--C--cycle);
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label("$B$",B,S);
 
label("$B$",B,S);
 
label("$C$",C,S);
 
label("$C$",C,S);
−
label("$E$",E,S);
+
label("$E$",E,NW);
−
label("$D$",D,S);
+
label("$D$",D,NE);
 
label("$13$",A--B,NW);
 
label("$13$",A--B,NW);
 
label("$15$",A--C,NE);
 
label("$15$",A--C,NE);
 
label("$14$",B--C,S);
 
label("$14$",B--C,S);
 +
label("$6.5$",B--E,N);
 +
label("$7$",C--D,N);
 
</asy>
 
</asy>
  
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<math>A^2=s(s-a)(s-b)(s-c)\\\\A^2=21(21-14)(21-15)(21-13)\\\\\boxed{A=84}</math>
 
<math>A^2=s(s-a)(s-b)(s-c)\\\\A^2=21(21-14)(21-15)(21-13)\\\\\boxed{A=84}</math>
  
−
Knowing that D is the midpoint of BC, thus the area of <math>\Delta ADC=\Delta ABD</math>, which is half the area of <math>\Delta ABC</math>, so <math>\Delta ADC=\Delta ABD=42</math>.
+
Knowing that D is the midpoint of BC, then <math>BD=CD=7</math>.
  
 
By [[Angle Bisector Theorem]] we know that:
 
By [[Angle Bisector Theorem]] we know that:
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And, we can easily see that <math>DE=0.5</math>, so,
 
And, we can easily see that <math>DE=0.5</math>, so,
  
−
<math>\frac{A_{\Delta ADE}}{84}=\frac{0.5}{14}</math>
+
$\frac{A_{
  
−
<math>\boxed{A_{\Delta ADE}=3}</math>
+
== Video Solution ==
 +
https://youtu.be/A5oxZp3jem4
  
 
== See also ==
 
== See also ==

Latest revision as of 16:37, 17 February 2024

Problem

In triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Which of the following is closest to the area of the triangle $ADE$?

$\text {(A)}\ 2 \qquad \text {(B)}\ 2.5 \qquad \text {(C)}\ 3 \qquad \text {(D)}\ 3.5 \qquad \text {(E)}\ 4$

Solution 1

The answer is exactly $3$, choice $\mathrm{(C)}$. We can find the area of triangle $ADE$ by using the simple formula $\frac{bh}{2}$. Dropping an altitude from $A$, we see that it has length $12$ (we can split the large triangle into a $9-12-15$ and a $5-12-13$ triangle). Then we can apply the Angle Bisector Theorem on triangle $ABC$ to solve for $BE$. Solving $\frac{13}{BE}=\frac{15}{14-BE}$, we get that $BE=\frac{13}{2}$. $D$ is the midpoint of $BC$ so $BD=7$. Thus we get the base of triangle $ADE, DE$, to be $\frac{1}{2}$ units long. Applying the formula $\frac{bh}{2}$, we get $\frac{12*\frac{1}{2}}{2}=3$.

Solution 2

The area of $ADE$ is $\frac{DE\cdot h}{2}=\frac{DE}{BC} \cdot \frac{BC\cdot h}{2}=\frac{DE}{BC}[ABC]$ where $h$ is the height of triangle $ABC$. Using Angle Bisector Theorem, we find $\frac{13}{BE}=\frac{15}{14-BE}$, which we solve to get $BE=\frac{13}{2}$. $D$ is the midpoint of $BC$ so $BD=7$. Thus we get the base of triangle $ADE, DE$, to be $\frac{1}{2}$ units long. We can now use Heron's Formula on $ABC$. \[s=\frac{AB+BC+AC}{2}=21\] \[[ABC]=\sqrt{(s)(s-AB)(s-BC)(s-AC)}=\sqrt{(21)(8)(7)(6)}=84\] \[\frac{DE}{BC}[ABC]=\frac{\frac{1}{2}}{14}\cdot 84=3\] Therefore, the answer is $\mathrm{C}$.

Solution 3

[asy] pair A,B,C,D,E; B=(0,0); C=(14,0); A=intersectionpoint(arc(B,13,0,90),arc(C,15,90,180)); draw(A--B--C--cycle); D=(7,0); E=(6.5,0); draw(A--E); draw(A--D); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$E$",E,NW); label("$D$",D,NE); label("$13$",A--B,NW); label("$15$",A--C,NE); label("$14$",B--C,S); label("$6.5$",B--E,N); label("$7$",C--D,N); [/asy]

Let's find the area of $\Delta ABC$ by Heron,

$s=\frac{a+b+c}{2}\\\\s=\frac{14+15+13}{2}\to\boxed{s=21}$

Then,

$A^2=s(s-a)(s-b)(s-c)\\\\A^2=21(21-14)(21-15)(21-13)\\\\\boxed{A=84}$

Knowing that D is the midpoint of BC, then $BD=CD=7$.

By Angle Bisector Theorem we know that:

$\frac{13}{BE}=\frac{15}{CE}$

$\boxed{CE=14-BE}$

$\frac{13}{BE}=\frac{15}{14-BE}$

$\boxed{BE=6.5}\Rightarrow CE=7.5$

Also, we know that:

$\frac{A_{\Delta ADE}}{A_{\Delta ABC}}=\frac{ED}{BC}$

And, we can easily see that $DE=0.5$, so,

$\frac{A_{

Video Solution

https://youtu.be/A5oxZp3jem4

See also

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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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