Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 AMC 12B Problems/Problem 11"

m (→‎Solution 4 (Heron's Formula, Pythagorean Theorem, Similar Triangles): Emphasized that we applied the Pytha. Thm. on right triangles.)
 
(16 intermediate revisions by 4 users not shown)
Line 6: Line 6:
 
==Diagram==
 
==Diagram==
 
<asy>
 
<asy>
−
size(8cm);
+
/* Made by Brendanb4321; edited by MRENTHUSIASM */
 +
size(250);
 
pair A = (5,12);
 
pair A = (5,12);
 
pair B = (0,0);
 
pair B = (0,0);
Line 13: Line 14:
 
pair D = 3/2*P;
 
pair D = 3/2*P;
 
pair E = 3*P;
 
pair E = 3*P;
−
draw(A--B--C--A);
+
draw(A--B--C--cycle^^A--D^^C--E--B);
−
draw(A--D);
 
−
draw(C--E--B);
 
  
−
dot("$A$",A,N);
+
dot("$A$",A,1.5*N);
−
dot("$B$",B,W);
+
dot("$B$",B,1.5*SW);
−
dot("$C$",C,ESE);
+
dot("$C$",C,1.5*SE);
−
dot("$D$",D,N);
+
dot("$D$",D,1.5*N);
−
dot("$P$",P,W);
+
dot("$P$",P,1.5*W);
−
dot("$E$",E,N);
+
dot("$E$",E,1.5*N);
  
−
defaultpen(fontsize(9pt));
+
label("$13$", (A+B)/2, 1.5*NW);
−
label("$13$", (A+B)/2, NW);
+
label("$14$", (B+C)/2, 1.5*S);
−
label("$14$", (B+C)/2, S);
+
label("$5$", (A+P)/2, NE);
−
label("$5$",(A+P)/2, NE);
 
 
label("$10$", (C+P)/2, NE);
 
label("$10$", (C+P)/2, NE);
 
</asy>
 
</asy>
 +
~Brendanb4321
  
−
==Solution 1 (analytic geometry)==
+
==Solution 1 (Analytic Geometry)==
 
Toss on the Cartesian plane with <math>A=(5, 12), B=(0, 0),</math> and <math>C=(14, 0)</math>. Then <math>\overline{AD}\parallel\overline{BC}, \overline{AB}\parallel\overline{CE}</math> by the trapezoid condition, where <math>D, E\in\overline{BP}</math>. Since <math>PC=10</math>, point <math>P</math> is <math>\tfrac{10}{15}=\tfrac{2}{3}</math> of the way from <math>C</math> to <math>A</math> and is located at <math>(8, 8)</math>. Thus line <math>BP</math> has equation <math>y=x</math>. Since <math>\overline{AD}\parallel\overline{BC}</math> and <math>\overline{BC}</math> is parallel to the ground, we know <math>D</math> has the same <math>y</math>-coordinate as <math>A</math>, except it'll also lie on the line <math>y=x</math>. Therefore, <math>D=(12, 12). \, \blacksquare</math>
 
Toss on the Cartesian plane with <math>A=(5, 12), B=(0, 0),</math> and <math>C=(14, 0)</math>. Then <math>\overline{AD}\parallel\overline{BC}, \overline{AB}\parallel\overline{CE}</math> by the trapezoid condition, where <math>D, E\in\overline{BP}</math>. Since <math>PC=10</math>, point <math>P</math> is <math>\tfrac{10}{15}=\tfrac{2}{3}</math> of the way from <math>C</math> to <math>A</math> and is located at <math>(8, 8)</math>. Thus line <math>BP</math> has equation <math>y=x</math>. Since <math>\overline{AD}\parallel\overline{BC}</math> and <math>\overline{BC}</math> is parallel to the ground, we know <math>D</math> has the same <math>y</math>-coordinate as <math>A</math>, except it'll also lie on the line <math>y=x</math>. Therefore, <math>D=(12, 12). \, \blacksquare</math>
  
 
To find the location of point <math>E</math>, we need to find the intersection of <math>y=x</math> with a line parallel to <math>\overline{AB}</math> passing through <math>C</math>. The slope of this line is the same as the slope of <math>\overline{AB}</math>, or <math>\tfrac{12}{5}</math>, and has equation <math>y=\tfrac{12}{5}x-\tfrac{168}{5}</math>. The intersection of this line with <math>y=x</math> is <math>(24, 24)</math>. Therefore point <math>E</math> is located at <math>(24, 24). \, \blacksquare</math>
 
To find the location of point <math>E</math>, we need to find the intersection of <math>y=x</math> with a line parallel to <math>\overline{AB}</math> passing through <math>C</math>. The slope of this line is the same as the slope of <math>\overline{AB}</math>, or <math>\tfrac{12}{5}</math>, and has equation <math>y=\tfrac{12}{5}x-\tfrac{168}{5}</math>. The intersection of this line with <math>y=x</math> is <math>(24, 24)</math>. Therefore point <math>E</math> is located at <math>(24, 24). \, \blacksquare</math>
  
−
The distance <math>DE</math> is equal to the distance between <math>(12, 12)</math> and <math>(24, 24)</math>, which is <math>\boxed{\textbf{(D)} ~12\sqrt{2}}</math>
+
The distance <math>DE</math> is equal to the distance between <math>(12, 12)</math> and <math>(24, 24)</math>, which is <math>\boxed{\textbf{(D) }12\sqrt2}</math>.
  
−
==Solution 2==
+
==Solution 2 (Stewart's Theorem)==
−
Using Stewart's Theorem we find <math>BP = 8\sqrt{2}</math>. From the similar triangles <math>BPA\sim DPC</math> and <math>BPC\sim EPA</math> we have
+
Using Stewart's Theorem we find <math>BP = 8\sqrt{2}</math>. From the similar triangles <math>BPA\sim DPC</math> and <math>BPC\sim EPA</math>, we have
−
<cmath>DP = BP\cdot\frac{PC}{PA} = 2BP</cmath>
+
<cmath>DP = BP\cdot\frac{PC}{PA} = \frac{1}{2}BP</cmath>
−
<cmath>EP = BP\cdot\frac{PA}{PC} = \frac12 BP</cmath>
+
<cmath>EP = BP\cdot\frac{PA}{PC} = 2BP</cmath>
−
So
+
So, <cmath>DE = \frac{3}{2}BP = \boxed{\textbf{(D) }12\sqrt2}.</cmath>
−
<cmath>DE = \frac{3}{2}BP = \boxed{\textbf{(D) }12\sqrt2}</cmath>
 
  
 
==Solution 3==
 
==Solution 3==
Line 61: Line 59:
 
&= 512
 
&= 512
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
And <math>x = 16\sqrt2</math>. The answer is then <math>\frac34x = \boxed{\textbf{(D) }12\sqrt2}</math>
+
And <math>x = 16\sqrt2</math>. The answer is then <math>\frac34x = \boxed{\textbf{(D) }12\sqrt2}</math>.
  
 
==Solution 4 (Heron's Formula, Pythagorean Theorem, Similar Triangles)==
 
==Solution 4 (Heron's Formula, Pythagorean Theorem, Similar Triangles)==
Line 75: Line 73:
 
It follows that the height of <math>ABCD</math> is <math>\frac{2[ABC]}{14}=12.</math>
 
It follows that the height of <math>ABCD</math> is <math>\frac{2[ABC]}{14}=12.</math>
  
−
Next, we drop the altitudes <math>\overline{AF}</math> and <math>\overline{DG}</math> of <math>ABCD.</math> By the Pythagorean Theorem on right <math>\triangle AFB,</math> we get <math>BF=5.</math> By the AA Similarity, <math>\triangle ADP\sim\triangle CBP,</math> with the ratio of similitude <math>1:2.</math> It follows that <math>AD=7.</math> Since <math>ADGF</math> is a rectangle, <math>FG=AD=7.</math> By the Pythagorean Theorem on right <math>\triangle DGB,</math> we get <math>BD=12\sqrt2.</math>
+
Next, we drop the altitudes <math>\overline{AF}</math> and <math>\overline{DG}</math> of <math>ABCD.</math> By the Pythagorean Theorem on right <math>\triangle AFB,</math> we get <math>BF=5.</math> By the AA Similarity, we have <math>\triangle ADP\sim\triangle CBP,</math> with the ratio of similitude <math>1:2.</math> It follows that <math>AD=7.</math> Since <math>ADGF</math> is a rectangle, we get <math>FG=AD=7.</math> By the Pythagorean Theorem on right <math>\triangle DGB,</math> we get <math>BD=12\sqrt2.</math>
  
−
By <math>\triangle ADP\sim\triangle CBP</math> again, we have <math>BP=8\sqrt2</math> and <math>DP=4\sqrt2.</math> Also, by the AA Similarity, <math>\triangle ABP\sim\triangle CEP,</math> with the ratio of similitude <math>1:2.</math> It follows that <math>EP=16\sqrt2.</math>
+
By <math>\triangle ADP\sim\triangle CBP</math> again, we get <math>BP=8\sqrt2</math> and <math>DP=4\sqrt2.</math> Also, by the AA Similarity, we have <math>\triangle ABP\sim\triangle CEP,</math> with the ratio of similitude <math>1:2.</math> It follows that <math>EP=16\sqrt2.</math>
  
−
Finally, <math>DE=EP-DP=\boxed{\textbf{(D) }12\sqrt2}.</math>
+
Finally, we obtain <math>DE=EP-DP=\boxed{\textbf{(D) }12\sqrt2}.</math>
  
−
<b>PS: If you memorize that the area of a <math>\mathbf{13}\textbf{-}\mathbf{14}\textbf{-}\mathbf{15}</math> triangle is <math>\mathbf{84,}</math> then the Heron's Formula part will be done instantly.</b>
+
<u><b>Remark</b></u>
 +
 
 +
It is well known that the area of a <math>13\text{-}14\text{-}15</math> triangle is <math>84.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
==Solution 5 (Barycentric coordinates)==
+
==Solution 5 (Barycentric Coordinates)==
  
 
(For those unfamiliar with barycentric coordinates, consider reading the barycentric coordinates article written by Evan Chen and Max Schindler here: https://artofproblemsolving.com/wiki/index.php/Resources_for_mathematics_competitions#Articles)
 
(For those unfamiliar with barycentric coordinates, consider reading the barycentric coordinates article written by Evan Chen and Max Schindler here: https://artofproblemsolving.com/wiki/index.php/Resources_for_mathematics_competitions#Articles)
  
 
We can find <math>P</math> in barycentric coordinates as <math>\Bigr(\frac{2}{3},0,\frac{1}{3}\Bigr)</math>. We can then write <math>\overline{BP}</math> as <math>x-2z=0</math>, where <math>(x,y,z)</math> defines a point in barycentric coordinates. We have <math>\overline{AD}\parallel \overline{BC}</math> as <math>y+z=0</math> and <math>\overline{CE}\parallel \overline{AB}</math> as <math>x+y=0</math>. We can then compute <math>D</math> and <math>E</math> by intersecting lines:
 
We can find <math>P</math> in barycentric coordinates as <math>\Bigr(\frac{2}{3},0,\frac{1}{3}\Bigr)</math>. We can then write <math>\overline{BP}</math> as <math>x-2z=0</math>, where <math>(x,y,z)</math> defines a point in barycentric coordinates. We have <math>\overline{AD}\parallel \overline{BC}</math> as <math>y+z=0</math> and <math>\overline{CE}\parallel \overline{AB}</math> as <math>x+y=0</math>. We can then compute <math>D</math> and <math>E</math> by intersecting lines:
−
 
 
<cmath>\begin{cases}
 
<cmath>\begin{cases}
 
x-2z=0\\
 
x-2z=0\\
Line 96: Line 95:
 
x+y+z=1
 
x+y+z=1
 
\end{cases}</cmath>
 
\end{cases}</cmath>
−
 
+
Which gives us <math>D=\left(1, -\frac{1}{2}, \frac{1}{2}\right)</math>. We can get <math>E</math> with:
−
Which gives us <math>D=(1, -\frac{1}{2}, \frac{1}{2})</math>. We can get <math>E</math> with:
 
−
 
 
 
<cmath>
 
<cmath>
 
\begin{cases}
 
\begin{cases}
Line 106: Line 103:
 
\end{cases}
 
\end{cases}
 
</cmath>
 
</cmath>
−
 
+
Which gives us <math>E=(2, -2, 1)</math>. Then, finding the displacement vector, we have <math>\overrightarrow{ED}=\left(1,-\frac{3}{2},\frac{1}{2}\right)</math>. Using the barycentric distance formula:
−
Which gives us <math>E=(2, -2, 1)</math>. Then, finding the displacement vector, we have <math>\overrightarrow{ED}=(1,-\frac{3}{2},\frac{1}{2})</math>. Using the barycentric distance formula:
+
<cmath>\begin{align*}
−
 
+
\text{dist}(D,E)&=\sqrt{-a^2yx-b^2zx-c^2xy} \\
−
<cmath>\text{dist}(D,E)=\sqrt{-a^2yx-b^2zx-c^2xy}</cmath>
+
&=\sqrt{-14^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)-15^2\Bigr(1\Bigr)\Bigr(\frac{1}{2}\Bigr)-13^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)} \\
−
<cmath>=\sqrt{-14^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)-15^2\Bigr(1\Bigr)\Bigr(\frac{1}{2}\Bigr)-13^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)}</cmath>
+
&=\boxed{\textbf{(D) }12\sqrt2}.
−
We get <math>\boxed{\textbf{(D)}12\sqrt{2}}</math>
+
\end{align*}</cmath>
  
 
==Video Solution by Punxsutawney Phil==
 
==Video Solution by Punxsutawney Phil==
 
https://YouTube.com/watch?v=yxt8-rUUosI&t=450s
 
https://YouTube.com/watch?v=yxt8-rUUosI&t=450s
  
−
== Video Solution by OmegaLearn (Using properties of 13-14-15 triangle) ==
+
== Video Solution by OmegaLearn (Properties of 13-14-15 Triangle) ==
 
https://youtu.be/mTcdKf5-FWg
 
https://youtu.be/mTcdKf5-FWg
  

Latest revision as of 19:40, 21 October 2024

Problem

Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$

$\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18$

Diagram

[asy] /* Made by Brendanb4321; edited by MRENTHUSIASM */ size(250); pair A = (5,12); pair B = (0,0); pair C = (14,0); pair P = 2/3*A+1/3*C; pair D = 3/2*P; pair E = 3*P; draw(A--B--C--cycle^^A--D^^C--E--B); dot("$A$",A,1.5*N); dot("$B$",B,1.5*SW); dot("$C$",C,1.5*SE); dot("$D$",D,1.5*N); dot("$P$",P,1.5*W); dot("$E$",E,1.5*N); label("$13$", (A+B)/2, 1.5*NW); label("$14$", (B+C)/2, 1.5*S); label("$5$", (A+P)/2, NE); label("$10$", (C+P)/2, NE); [/asy] ~Brendanb4321

Solution 1 (Analytic Geometry)

Toss on the Cartesian plane with $A=(5, 12), B=(0, 0),$ and $C=(14, 0)$. Then $\overline{AD}\parallel\overline{BC}, \overline{AB}\parallel\overline{CE}$ by the trapezoid condition, where $D, E\in\overline{BP}$. Since $PC=10$, point $P$ is $\tfrac{10}{15}=\tfrac{2}{3}$ of the way from $C$ to $A$ and is located at $(8, 8)$. Thus line $BP$ has equation $y=x$. Since $\overline{AD}\parallel\overline{BC}$ and $\overline{BC}$ is parallel to the ground, we know $D$ has the same $y$-coordinate as $A$, except it'll also lie on the line $y=x$. Therefore, $D=(12, 12). \, \blacksquare$

To find the location of point $E$, we need to find the intersection of $y=x$ with a line parallel to $\overline{AB}$ passing through $C$. The slope of this line is the same as the slope of $\overline{AB}$, or $\tfrac{12}{5}$, and has equation $y=\tfrac{12}{5}x-\tfrac{168}{5}$. The intersection of this line with $y=x$ is $(24, 24)$. Therefore point $E$ is located at $(24, 24). \, \blacksquare$

The distance $DE$ is equal to the distance between $(12, 12)$ and $(24, 24)$, which is $\boxed{\textbf{(D) }12\sqrt2}$.

Solution 2 (Stewart's Theorem)

Using Stewart's Theorem we find $BP = 8\sqrt{2}$. From the similar triangles $BPA\sim DPC$ and $BPC\sim EPA$, we have \[DP = BP\cdot\frac{PC}{PA} = \frac{1}{2}BP\] \[EP = BP\cdot\frac{PA}{PC} = 2BP\] So, \[DE = \frac{3}{2}BP = \boxed{\textbf{(D) }12\sqrt2}.\]

Solution 3

Let $x$ be the length $PE$. From the similar triangles $BPA\sim DPC$ and $BPC\sim EPA$ we have \[BP = \frac{PA}{PC}x = \frac12 x\] \[PD = \frac{PA}{PC}BP = \frac14 x\] Therefore $BD = DE = \frac{3}{4}x$. Now extend line $CD$ to the point $Z$ on $AE$, forming parallelogram $ZABC$. As $BD = DE$ we also have $EZ = ZC = 13$ so $EC = 26$.

We now use the Law of Cosines to find $x$ (the length of $PE$): \[x^2 = EC^2 + PC^2 - 2(EC)(PC)\cos{(PCE)} = 26^2 + 10^2 - 2\cdot 26\cdot 10\cos(\angle PCE)\] As $\angle PCE = \angle BAC$, we have (by Law of Cosines on triangle $BAC$) \[\cos(\angle PCE) = \frac{13^2 + 15^2 - 14^2}{2\cdot 13\cdot 15}.\] Therefore \begin{align*} x^2 &= 26^2 + 10^2 - 2\cdot 26\cdot 10\cdot\frac{198}{2\cdot 13\cdot 15}\\ &= 776 - 264\\ &= 512 \end{align*} And $x = 16\sqrt2$. The answer is then $\frac34x = \boxed{\textbf{(D) }12\sqrt2}$.

Solution 4 (Heron's Formula, Pythagorean Theorem, Similar Triangles)

Let the brackets denote areas. By Heron's Formula, we have \begin{align*} [ABC]&=\sqrt{\frac{13+14+15}{2}\left(\frac{13+14+15}{2}-13\right)\left(\frac{13+14+15}{2}-14\right)\left(\frac{13+14+15}{2}-15\right)} \\ &=\sqrt{21\left(21-13\right)\left(21-14\right)\left(21-15\right)} \\ &=\sqrt{21\left(8\right)\left(7\right)\left(6\right)} \\ &=\sqrt{\left(3\cdot7\right)\left(2^3\right)\left(7\right)\left(2\cdot3\right)} \\ &=2^2\cdot3\cdot7 \\ &=84. \end{align*} It follows that the height of $ABCD$ is $\frac{2[ABC]}{14}=12.$

Next, we drop the altitudes $\overline{AF}$ and $\overline{DG}$ of $ABCD.$ By the Pythagorean Theorem on right $\triangle AFB,$ we get $BF=5.$ By the AA Similarity, we have $\triangle ADP\sim\triangle CBP,$ with the ratio of similitude $1:2.$ It follows that $AD=7.$ Since $ADGF$ is a rectangle, we get $FG=AD=7.$ By the Pythagorean Theorem on right $\triangle DGB,$ we get $BD=12\sqrt2.$

By $\triangle ADP\sim\triangle CBP$ again, we get $BP=8\sqrt2$ and $DP=4\sqrt2.$ Also, by the AA Similarity, we have $\triangle ABP\sim\triangle CEP,$ with the ratio of similitude $1:2.$ It follows that $EP=16\sqrt2.$

Finally, we obtain $DE=EP-DP=\boxed{\textbf{(D) }12\sqrt2}.$

Remark

It is well known that the area of a $13\text{-}14\text{-}15$ triangle is $84.$

~MRENTHUSIASM

Solution 5 (Barycentric Coordinates)

(For those unfamiliar with barycentric coordinates, consider reading the barycentric coordinates article written by Evan Chen and Max Schindler here: https://artofproblemsolving.com/wiki/index.php/Resources_for_mathematics_competitions#Articles)

We can find $P$ in barycentric coordinates as $\Bigr(\frac{2}{3},0,\frac{1}{3}\Bigr)$. We can then write $\overline{BP}$ as $x-2z=0$, where $(x,y,z)$ defines a point in barycentric coordinates. We have $\overline{AD}\parallel \overline{BC}$ as $y+z=0$ and $\overline{CE}\parallel \overline{AB}$ as $x+y=0$. We can then compute $D$ and $E$ by intersecting lines: \[\begin{cases} x-2z=0\\ y+z=0\\ x+y+z=1 \end{cases}\] Which gives us $D=\left(1, -\frac{1}{2}, \frac{1}{2}\right)$. We can get $E$ with: \[\begin{cases} x-2z=0\\ x+y=0\\ x+y+z=1 \end{cases}\] Which gives us $E=(2, -2, 1)$. Then, finding the displacement vector, we have $\overrightarrow{ED}=\left(1,-\frac{3}{2},\frac{1}{2}\right)$. Using the barycentric distance formula: \begin{align*} \text{dist}(D,E)&=\sqrt{-a^2yx-b^2zx-c^2xy} \\ &=\sqrt{-14^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)-15^2\Bigr(1\Bigr)\Bigr(\frac{1}{2}\Bigr)-13^2\Bigr(-\frac{3}{2}\Bigr)\Bigr(\frac{1}{2}\Bigr)} \\ &=\boxed{\textbf{(D) }12\sqrt2}. \end{align*}

Video Solution by Punxsutawney Phil

https://YouTube.com/watch?v=yxt8-rUUosI&t=450s

Video Solution by OmegaLearn (Properties of 13-14-15 Triangle)

https://youtu.be/mTcdKf5-FWg

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

See Also

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_11&oldid=230299"