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Difference between revisions of "2021 AMC 12A Problems/Problem 17"

m (→‎Solution 3 formatting)
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Angle chasing reveals that <math>\triangle BPC\sim\triangle BDA</math>, therefore  
 
Angle chasing reveals that <math>\triangle BPC\sim\triangle BDA</math>, therefore  
−
<cmath>2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43}</cmath>
+
<cmath>2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43},</cmath>
−
<cmath>AB=86</cmath>
+
or <math>AB=86</math>.
 +
 
 
Additional angle chasing shows that <math>\triangle ABO \sim\triangle CDO</math>, therefore
 
Additional angle chasing shows that <math>\triangle ABO \sim\triangle CDO</math>, therefore
−
<cmath>2=\frac{AB}{CD}=\frac{BO}{OD}=\frac{BP+11}{BP-11}</cmath>
+
<cmath>2=\frac{AB}{CD}=\frac{BO}{OD}=\frac{BP+11}{BP-11},</cmath>
−
<cmath>BP=33 \Rightarrow BD=66</cmath>
+
or <math>BP=33</math> and <math>BD=66</math>.
 +
 
 
Since <math>\triangle ADB</math> is right, the Pythagorean theorem implies that
 
Since <math>\triangle ADB</math> is right, the Pythagorean theorem implies that
−
<cmath>AD=\sqrt{86^2-66^2}</cmath>
+
<cmath>AD=\sqrt{86^2-66^2}=4\sqrt{190}.</cmath>
−
<cmath>AD=4\sqrt{190}</cmath>
+
The answer is <math>4+190=\boxed{\textbf{(D) }194}</math>.
−
<math>4\sqrt{190}\implies 4 + 190 = \boxed{\textbf{D) } 194}</math>
 
  
 
~mn28407 (minor edits by eagleye)
 
~mn28407 (minor edits by eagleye)
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(2) Applying Pythagorean Theorem on <math>\triangle CDP</math> gives <math>(11+x)^2 + y^2 = 43^2.</math>
 
(2) Applying Pythagorean Theorem on <math>\triangle CDP</math> gives <math>(11+x)^2 + y^2 = 43^2.</math>
  
−
(3) <math>\triangle BPC \sim \triangle BDA</math> with ratio <math>1:2,</math> so <math>AD = 2y.</math> (Using the fact that <math>P</math> is the midpoint of <math>BD</math>)
+
(3) <math>\triangle BPC \sim \triangle BDA</math> with ratio <math>1:2,</math> so <math>AD = 2y</math> using the fact that <math>P</math> is the midpoint of <math>BD</math>.
  
−
Thus, <math>\frac{xt}{11} = 2y,</math> or <math>x = 22.</math> And <math>y = \sqrt{43^2 - 33^2} = 2 \sqrt{190},</math> so <math>AD = 4 \sqrt{190}</math> and the answer is <math>\boxed{194}.</math>
+
Thus, <math>\frac{xt}{11} = 2y,</math> or <math>x = 22.</math> And <math>y = \sqrt{43^2 - 33^2} = 2 \sqrt{190},</math> so <math>AD = 4 \sqrt{190}</math> and the answer is <math>4+190=\boxed{\textbf{(D) }194}.</math>
  
 
~ ccx09
 
~ ccx09
Line 60: Line 61:
 
Therefore, <math>\overline{AC}</math> and <math>\overline{BD}</math> are both medians of <math>\triangle ABE</math>. This means that <math>O</math> is the centroid of <math>\triangle ABE</math>; therefore, because the centroid divides the median in a 2:1 ratio, <math>\frac{BO}{2} = DO = \frac{BD}{3}</math>. Recall that <math>P</math> is the midpoint of <math>BD</math>; <math>DP = \frac{BD}{2}</math>. The question tells us that <math>OP = 11</math>; <math>DP-DO=11</math>; we can write this in terms of <math>DB</math>; <math>\frac{DB}{2}-\frac{DB}{3} = \frac{DB}{6} = 11 \implies DB = 66</math>.
 
Therefore, <math>\overline{AC}</math> and <math>\overline{BD}</math> are both medians of <math>\triangle ABE</math>. This means that <math>O</math> is the centroid of <math>\triangle ABE</math>; therefore, because the centroid divides the median in a 2:1 ratio, <math>\frac{BO}{2} = DO = \frac{BD}{3}</math>. Recall that <math>P</math> is the midpoint of <math>BD</math>; <math>DP = \frac{BD}{2}</math>. The question tells us that <math>OP = 11</math>; <math>DP-DO=11</math>; we can write this in terms of <math>DB</math>; <math>\frac{DB}{2}-\frac{DB}{3} = \frac{DB}{6} = 11 \implies DB = 66</math>.
  
−
 
+
We are almost finished. Each side length of <math>\triangle ABD</math> is twice as long as the corresponding side length <math>\triangle CBP</math> or <math>\triangle CPD</math>, since those triangles are similar; this means that <math>AB = 2 \cdot 43 = 86</math>. Now, by Pythagorean theorem on <math>\triangle ABD</math>, <math>AB^{2} - BD^{2} = AD^{2} \implies 86^{2}-66^{2} = AD^{2} \implies AD = \sqrt{3040} \implies AD = 4 \sqrt{190}</math>. <math>4+190 = \boxed{\textbf{(D) }194}</math>
−
We are almost finished. Each side length of <math>\triangle ABD</math> is twice as long as the corresponding side length <math>\triangle CBP</math> or <math>\triangle CPD</math>, since those triangles are similar; this means that <math>AB = 2 \cdot 43 = 86</math>. Now, by Pythagorean theorem on <math>\triangle ABD</math>, <math>AB^{2} - BD^{2} = AD^{2} \implies 86^{2}-66^{2} = AD^{2} \implies AD = \sqrt{3040} \implies AD = 4 \sqrt{190}</math>. <math>4+190 = \boxed{194, \textbf{D}}</math>
 
  
 
~ ihatemath123
 
~ ihatemath123

Revision as of 14:11, 22 August 2021

The following problem is from both the 2021 AMC 10A #17 and 2021 AMC 12A #17, so both problems redirect to this page.

Problem

Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?

$\textbf{(A) }65 \qquad \textbf{(B) }132 \qquad \textbf{(C) }157 \qquad \textbf{(D) }194\qquad \textbf{(E) }215$

Diagram

600px

~MRENTHUSIASM (by Geometry Expressions)

Solution 1

Angle chasing reveals that $\triangle BPC\sim\triangle BDA$, therefore \[2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43},\] or $AB=86$.

Additional angle chasing shows that $\triangle ABO \sim\triangle CDO$, therefore \[2=\frac{AB}{CD}=\frac{BO}{OD}=\frac{BP+11}{BP-11},\] or $BP=33$ and $BD=66$.

Since $\triangle ADB$ is right, the Pythagorean theorem implies that \[AD=\sqrt{86^2-66^2}=4\sqrt{190}.\] The answer is $4+190=\boxed{\textbf{(D) }194}$.

~mn28407 (minor edits by eagleye)

Solution 2 (Similar Triangles, Areas, Pythagorean Theorem)

Since $\triangle BCD$ is isosceles with base $\overline{BD},$ it follows that median $\overline{CP}$ is also an altitude. Let $OD=x$ and $CP=h,$ so $PB=x+11.$

Since $\angle AOD=\angle COP$ by vertical angles, we conclude that $\triangle AOD\sim\triangle COP$ by AA, from which $\frac{AD}{CP}=\frac{OD}{OP},$ or \[AD=CP\cdot\frac{OD}{OP}=h\cdot\frac{x}{11}.\] Let the brackets denote areas. Notice that $[AOD]=[BOC]$ (By the same base and height, we deduce that $[ACD]=[BDC].$ Subtracting $[OCD]$ from both sides gives $[AOD]=[BOC].$). Doubling both sides produces \begin{align*} 2[AOD]&=2[BOC] \\ OD\cdot AD&=OB\cdot CP \\ x\left(\frac{hx}{11}\right)&=(x+22)h \\ x^2&=11(x+22). \end{align*} Rearranging and factoring result in $(x-22)(x+11)=0,$ from which $x=22.$

Applying the Pythagorean Theorem to right $\triangle CPB,$ we have \[h=\sqrt{43^2-33^2}=\sqrt{(43+33)(43-33)}=\sqrt{760}=2\sqrt{190}.\] Finally, we get \[AD=h\cdot\frac{x}{11}=4\sqrt{190},\] so the answer is $4+190=\boxed{\textbf{(D) }194}.$

~MRENTHUSIASM

Solution 3 (Short)

Let $CP = y$. $CP$ a is perpendicular bisector of $DB.$ Then, let $DO = x,$ thus $DP = PB = 11+x.$

(1) $\triangle CPO \sim \triangle ADO,$ so we get $\frac{AD}{x} = \frac{y}{11},$ or $AD = \frac{xy}{11}.$

(2) Applying Pythagorean Theorem on $\triangle CDP$ gives $(11+x)^2 + y^2 = 43^2.$

(3) $\triangle BPC \sim \triangle BDA$ with ratio $1:2,$ so $AD = 2y$ using the fact that $P$ is the midpoint of $BD$.

Thus, $\frac{xt}{11} = 2y,$ or $x = 22.$ And $y = \sqrt{43^2 - 33^2} = 2 \sqrt{190},$ so $AD = 4 \sqrt{190}$ and the answer is $4+190=\boxed{\textbf{(D) }194}.$

~ ccx09

Solution 4 (Extending the Line)

Observe that $\triangle BPC$ is congruent to $\triangle DPC$; both are similar to $\triangle BDA$. Let's extend $\overline{AD}$ and $\overline{BC}$ past points $D$ and $C$ respectively, such that they intersect at a point $E$. Observe that $\angle BDE$ is $90$ degrees, and that $\angle DBE \cong \angle PBC \cong \angle DBA \implies \angle DBE \cong \angle DBA$. Thus, by ASA, we know that $\triangle ABD \cong \triangle EBD$, thus, $AD = ED$, meaning $D$ is the midpoint of $AE$. Let $M$ be the midpoint of $\overline{DE}$. Note that $\triangle CME$ is congruent to $\triangle BPC$, thus $BC = CE$, meaning $C$ is the midpoint of $\overline{BE}.$

Therefore, $\overline{AC}$ and $\overline{BD}$ are both medians of $\triangle ABE$. This means that $O$ is the centroid of $\triangle ABE$; therefore, because the centroid divides the median in a 2:1 ratio, $\frac{BO}{2} = DO = \frac{BD}{3}$. Recall that $P$ is the midpoint of $BD$; $DP = \frac{BD}{2}$. The question tells us that $OP = 11$; $DP-DO=11$; we can write this in terms of $DB$; $\frac{DB}{2}-\frac{DB}{3} = \frac{DB}{6} = 11 \implies DB = 66$.

We are almost finished. Each side length of $\triangle ABD$ is twice as long as the corresponding side length $\triangle CBP$ or $\triangle CPD$, since those triangles are similar; this means that $AB = 2 \cdot 43 = 86$. Now, by Pythagorean theorem on $\triangle ABD$, $AB^{2} - BD^{2} = AD^{2} \implies 86^{2}-66^{2} = AD^{2} \implies AD = \sqrt{3040} \implies AD = 4 \sqrt{190}$. $4+190 = \boxed{\textbf{(D) }194}$

~ ihatemath123

Video Solution (Using Similar Triangles, Pythagorean Theorem)

https://youtu.be/gjeSGJy_ld4

~ pi_is_3.14

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=rtdovluzgQs

Video Solution by Mathematical Dexterity

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

See also

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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
2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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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