2022 AMC 10A Problems/Problem 13

Problem

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ What is $AD?$

$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

Diagram

$[asy] /* Made by MRENTHUSIASM */ size(300); real r = 4*sqrt(114)/13; pair A, B, C, D, P, X, Y; A = origin; B = (2,r); C = (3/2*sqrt(2^2+r^2),0); D = A + 2*(C-B); P = B + 2*dir(C-B); X = intersectionpoint(B--D,A--P); Y = intersectionpoint(B--D,A--C); dot("$A$",A,1.5*W,linewidth(4)); dot("$B$",B,1.5*N,linewidth(4)); dot("$C$",C,1.5*E,linewidth(4)); dot("$P$",P,1.5*dir(P),linewidth(4)); dot("$D$",D,1.5*dir(D),linewidth(4)); dot(X^^Y,linewidth(4)); markscalefactor=0.03; draw(rightanglemark(B,X,A),red); draw(anglemark(P,A,B,20), red); draw(anglemark(C,A,P,20), red); add(pathticks(anglemark(P,A,B,20), n = 1, r = 0.1, s = 7, red)); add(pathticks(anglemark(C,A,P,20), n = 1, r = 0.1, s = 7, red)); draw(A--B--C--cycle^^A--P^^B--D^^A--D); draw(B--C,MidArrow(0.3cm,Fill(red))); draw(A--D,MidArrow(0.3cm,Fill(red))); label("$2$",midpoint(B--P),rotate(90)*dir(midpoint(P--B)--P),red); label("$3$",midpoint(P--C),rotate(90)*dir(midpoint(C--P)--C),red); [/asy]$

~MRENTHUSIASM

Solution 1 (The extra line)

Let the intersection of $AC$ and $BD$ be $M$ , and the intersection of $AP$ and $BD$ be $N$ . Draw a line from $M$ to $BC$ , and label the point $O$ . We have $\triangle BPN \sim \triangle BOM$ , with a ratio of $2$ , so $BO = 4$ and $OC = 1$ . We also have $\triangle COM \sim \triangle CAP$ with ratio $3$ .

Suppose the area of $\triangle COM$ is $x$ . Then, $[\triangle CAP] = 9x$ . Because $\triangle CAP$ and $\triangle BAP$ share the same height and have a base ratio of $3:2$ , $[\triangle BAP] = 6x$ . Because $\triangle BOM$ and $\triangle COM$ share the same height and have a base ratio of $4:1$ , $[\triangle BOM] = 4x$ , $[\triangle BPN] = x$ , and $[OMNP] = 4x - x = 3x$ . Thus, $[\triangle MAN] = [\triangle BAN] = 5x$ . Finally, because $BPN \sim ADN$ and the ratio $AN : PN$ is $5$ (because $[\triangle BAN] : [\triangle BPN] = 5$ and they share a side), $AD = 2 \cdot 5 = \boxed{\textbf{(C) } 10}.

~mathboy100

==Solution 2 (Generalization)==

Suppose that$ (Error compiling LaTeX. Unknown error_msg)\overline{BD} $intersect$ \overline{AP} $and$ \overline{AC} $at$ X $and$ Y, $respectively. By Angle-Side-Angle, we conclude that$ \triangle ABX\cong\triangle AYX. $Let$ AB=AY=2x. $By the Angle Bisector Theorem, we have$ AC=3x, $or$ YC=x. $By alternate interior angles, we get$ \angle YAD=\angle YCB $and$ \angle YDA=\angle YBC. $Note that$ \triangle ADY \sim \triangle CBY $by the Angle-Angle Similarity, with the ratio of similitude$ \frac{AY}{CY}=2. $It follows that$ AD=2CB=2(BP+PC)=\boxed{\textbf{(C) } 10}.$

~MRENTHUSIASM

Solution 3 (Assumption)

$[asy] size(300); pair A, B, C, P, XX, D; B = (0,0); P = (2,0); C = (5,0); A=(0,4.47214); D = A + (10,0); draw(A--B--C--cycle, linewidth(1)); dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, E); dot("$P$", P, S); dot("$D$", D, E); markscalefactor = 0.1; draw(anglemark(B,A,P)); markscalefactor = 0.12; draw(anglemark(P,A,C)); draw(P--A--D--B, linewidth(1)); XX = intersectionpoints(A--P,B--D)[0]; dot("$X$", XX, dir(150)); markscalefactor = 0.03; draw(rightanglemark(A,B,C)); draw(rightanglemark(D,XX,P)); [/asy]$ Since there is only one possible value of $AD$ , we assume $\angle{B}=90^{\circ}$ . By the angle bisector theorem, $\frac{AB}{AC}=\frac{2}{3}$ , so $AB=2\sqrt{5}$ and $AC=3\sqrt{5}$ . Now observe that $\angle{BAD}=90^{\circ}$ . Let the intersection of $BD$ and $AP$ be $X$ . Then $\angle{ABD}=90^{\circ}-\angle{BAX}=\angle{APB}$ . Consequently, $\bigtriangleup DAB \sim \bigtriangleup ABP$ and therefore $\frac{DA}{AB} = \frac{AB}{BP}$ , so $AD=\boxed{\textbf{(C) }10}$ , and we're done!

~Bxiao31415

Video Solution 1

https://youtu.be/_0_EGdkhOFg

- Whiz

Video Solution by OmegaLearn

https://youtu.be/77JIN0iVizA

~ pi_is_3.14

Video Solution (Quick and Simple)

https://youtu.be/m1-7E8T_i_E

~Education, the Study of Everything

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AMC 10 Problems and Solutions

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