Difference between revisions of "2022 AMC 10A Problems/Problem 13"

Revision as of 01:20, 26 November 2022

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 $\overline{BD}$ 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 (Coord geo, slopes)

Let point $B$ be the origin, with $C$ being on the positive $x$ -axis and $A$ being in the first quadrant.

By the Angle Bisector Theorem, $AB:AC = 2:3$ . Thus, assume that $AB = 4$ , and $AC = 6$ .

Let the perpendicular from $A$ to $BC$ be $AM$ .

We have $[ABC] = \sqrt{\frac{15}{2}\left(\frac{15}{2}-4\right)\left(\frac{15}{2}-5\right)\left(\frac{15}{2}-6\right)} = \frac{15}{4}\sqrt{7}.$

Hence, $AM = \frac{\frac{15}{4}\sqrt{7}}{\frac{5}{2}} = \frac{3}{2}\sqrt{7}.$

Next, we have $BM^2 + AM^2 = AB^2$ $\therefore BM = \sqrt{16 - \frac{63}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \textrm{ and } PM = \frac{3}{2}.$

The slope of line $AP$ is thus $\frac{-\frac{3}{2}\sqrt{7}}{\frac{3}{2}} = -\sqrt{7}.$

Therefore, since the slopes of perpendicular lines have a product of $-1$ , the slope of line $BD$ is $\frac{1}{\sqrt{7}}$ . This means that we can solve for the coordinates of $D$ :

$y = \frac{3}{2}\sqrt{7}$ $y = \frac{1}{\sqrt{7}}x$ $\frac{1}{\sqrt{7}}x = \frac{3}{2}\sqrt{7}$ $x = \frac{7 \cdot 3}{2} = \frac{21}{2}$ $D = \left(\frac{21}{2}, \frac{3}{2}\sqrt{7}\right).$

We also know that the coordinates of $A$ are $\left(\frac{1}{2}, \frac{3}{2}\sqrt{7}\right)$ , because $BM = \frac{1}{2}$ and $AM = \frac{3}{2}\sqrt{7}$ .

Since the $y$ -coordinates of $A$ and $D$ are the same, and their $x$ -coordinates differ by $10$ , the distance between them is $10$ . Our answer is $\boxed{\textbf{(C) }10}.$

~mathboy100

Solution 4 (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

@@ Line 68: / Line 68: @@
 Next, we have <cmath>BM^2 + AM^2 = AB^2</cmath>
-<cmath>\therefore BM = \sqrt{16 - \frac{63}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \textrm{ and } PM = \frac{1}{2}.</cmath>
+<cmath>\therefore BM = \sqrt{16 - \frac{63}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \textrm{ and } PM = \frac{3}{2}.</cmath>
+The slope of line <math>AP</math> is thus <cmath>\frac{-\frac{3}{2}\sqrt{7}}{\frac{3}{2}} = -\sqrt{7}.</cmath>
+Therefore, since the slopes of perpendicular lines have a product of <math>-1</math>, the slope of line <math>BD</math> is <math>\frac{1}{\sqrt{7}}</math>. This means that we can solve for the coordinates of <math>D</math>:
+<cmath>y = \frac{3}{2}\sqrt{7}</cmath>
+<cmath>y = \frac{1}{\sqrt{7}}x</cmath>
+<cmath>\frac{1}{\sqrt{7}}x = \frac{3}{2}\sqrt{7}</cmath>
+<cmath>x = \frac{7 \cdot 3}{2} = \frac{21}{2}</cmath>
+<cmath>D = \left(\frac{21}{2}, \frac{3}{2}\sqrt{7}\right).</cmath>
+We also know that the coordinates of <math>A</math> are <math>\left(\frac{1}{2}, \frac{3}{2}\sqrt{7}\right)</math>, because <math>BM = \frac{1}{2}</math> and <math>AM = \frac{3}{2}\sqrt{7}</math>.
+Since the <math>y</math>-coordinates of <math>A</math> and <math>D</math> are the same, and their <math>x</math>-coordinates differ by <math>10</math>, the distance between them is <math>10</math>. Our answer is <math>\boxed{\textbf{(C) }10}.</math>
+ ~mathboy100
 == Solution 4 (Assumption) ==

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