2021 AMC 12B Problems/Problem 24

Problem

Let $ABCD$ be a parallelogram with area $15$ . Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

$[asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy]$

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$

$\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113$

Solution 1

simple Video Solution Using trigonometry and Equations

https://youtu.be/ZB-VN02H6mU ~hippopotamus1

Solution

Let $X$ denote the intersection point of the diagonals $AC$ and $BD$ . Remark that by symmetry $X$ is the midpoint of both $\overline{PQ}$ and $\overline{RS}$ , so $XP = XQ = 3$ and $XR = XS = 4$ . Now note that since $\angle APB = \angle ARB = 90^\circ$ , quadrilateral $ARPB$ is cyclic, and so $XR\cdot XA = XP\cdot XB,$ which implies $\tfrac{XA}{XB} = \tfrac{XP}{XR} = \tfrac34$ .

Thus let $x> 0$ be such that $XA = 3x$ and $XB = 4x$ . Then Pythagorean Theorem on $\triangle APX$ yields $AP = \sqrt{AX^2 - XP^2} = 3\sqrt{x^2-1}$ , and so $[ABCD] = 2[ABD] = AP\cdot BD = 3\sqrt{x^2-1}\cdot 8x = 24x\sqrt{x^2-1}.$ Solving this for $x^2$ yields $x^2 = \tfrac12 + \tfrac{\sqrt{41}}8$ , and so $(8x)^2 = 64x^2 = 64\left(\tfrac12 + \tfrac{\sqrt{41}}8\right) = 32 + 8\sqrt{41}.$ The requested answer is $32 + 8 + 41 = \boxed{81}$ .

Solution 2(Trig)

Let $X$ denote the intersection point of the diagonals $AC$ and $BD,$ and let $\theta = \angle{COB}$ . Then, by the given conditions, $XR = 4,$ $XQ = 3,$ $[XCB] = \frac{15}{4}$ . So, $XC = \frac{3}{\cos \theta}$ $XB \cos \theta = 4$ $\frac{1}{2} XB XC \sin \theta = \frac{15}{4}$ Combining the above 3 equations, we get $\frac{\sin \theta }{\cos^2 \theta} = \frac{5}{8}.$ Since we want to find $d^2 = 4XB^2 = \frac{64}{\cos^2 \theta},$ we let $x = \frac{1}{\cos^2 \theta}.$ Then $\frac{\sin^2 \theta }{\cos^4 \theta} = \frac{1-\cos ^2 \theta}{\cos^4 \theta} = x^2 - x = \frac{25}{64}.$ Solving this, we get $x = \frac{4 + \sqrt{41}}{8},$ so $d^2 = 64x = 32 + 8\sqrt{41}$ . $\boxed{81}$

Solution 3 (Similar Triangles and Algebra)

Let $X$ be the intersection of diagonals $AC$ and $BD$ . By symmetry $[\triangle XCB] = \frac{15}{4}$ , $XQ = 3$ and $XR = 4$ , so now we have reduced all of the conditions one quadrant. Let $CQ = x$ . $XC = \sqrt{x^2+9}$ , $RB = \frac{4x}{3}$ by similar triangles and using the area condition we get $\frac{4}{3} \cdot x \cdot \sqrt{x^2+9} = \frac{15}{2}$ . Note that it suffices to find $OB = \frac{4}{3}\sqrt{x^2+9}$ because we can double and square it to get $d^2$ . Solving for $a = x^2$ in the above equation, and then using $d^2 = \frac{64}{9}(x^2+9) = 8\sqrt{41} + 32 \Rightarrow \boxed{81}$ .

Solution 4 (Similar Triangles)

Again, Let $X$ be the intersection of diagonals $AC$ and $BD$ . Note that triangles $\triangle QXC$ and $\triangle BXR$ are similar because they are right triangles and share $\angle CXQ$ . First, call the length of $XB = \frac{d}{2}$ . By the definition of an area of a parallelogram, $CQ \cdot 2XB = 15$ , so $CQ = \frac{15}{d}$ . Using similar triangles on $\triangle QXC$ and $\triangle BXR$ , $\frac{CQ}{XQ} = \frac{BR}{XR}$ . Therefore, finding $BR$ , $BR = \frac{XR}{XQ} \cdot CQ = \frac{4}{3} \cdot \frac{15}{d} = \frac{20}{d}$ . Now, applying the Pythagorean theorem once, we find $(\frac{20}{d}) ^2$ + $(4)^2$ = $(\frac{d}{2}) ^2$ . Solving this equation for $d^2$ , we find $d^2=\frac{64+\sqrt{4096+6400}}{2}=32+8\sqrt{41} \Rightarrow \boxed{81}$ .

Video Solution by OmegaLearn (Cyclic Quadrilateral and Power of a Point)

https://youtu.be/1zhwR9B2Gy8

~ pi_is_3.14

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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