Difference between revisions of "1987 AIME Problems/Problem 15"

Revision as of 00:45, 27 December 2021

Problem

Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$ , as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$ .

Solution

Because all the triangles in the figure are similar to triangle $ABC$ , it's a good idea to use area ratios. In the diagram above, $\frac {T_1}{T_3} = \frac {T_2}{T_4} = \frac {441}{440}.$ Hence, $T_3 = \frac {440}{441}T_1$ and $T_4 = \frac {440}{441}T_2$ . Additionally, the area of triangle $ABC$ is equal to both $T_1 + T_2 + 441$ and $T_3 + T_4 + T_5 + 440.$

Setting the equations equal and solving for $T_5$ , $T_5 = 1 + T_1 - T_3 + T_2 - T_4 = 1 + \frac {T_1}{441} + \frac {T_2}{441}$ . Therefore, $441T_5 = 441 + T_1 + T_2$ . However, $441 + T_1 + T_2$ is equal to the area of triangle $ABC$ ! This means that the ratio between the areas $T_5$ and $ABC$ is $441$ , and the ratio between the sides is $\sqrt {441} = 21$ . As a result, $AB = 21\sqrt {440} = \sqrt {AC^2 + BC^2}$ . We now need $(AC)(BC)$ to find the value of $AC + BC$ , because $AB^2 + BC^2 + 2(AC)(BC) = (AC + BC)^2$ .

Let $h$ denote the height to the hypotenuse of triangle $ABC$ . Notice that $h - \frac {1}{21}h = \sqrt {440}$ . (The height of $ABC$ decreased by the corresponding height of $T_5$ ) Thus, $(AB)(h) = (AC)(BC) = 22\cdot 21^2$ . Because $AC^2 + BC^2 + 2(AC)(BC) = (AC + BC)^2 = 21^2\cdot22^2$ , $AC + BC = (21)(22) = \boxed{462}$ .

Easy Trig Solution

Let $\tan\angle ABC = x$ . Now using the 1st square, $AC=21(1+x)$ and $CB=21(1+x^{-1})$ . Using the second square, $AB=\sqrt{440}(1+x+x^{-1})$ . We have $AC^2+CB^2=AB^2$ , or $441(x^2+x^{-2}+2x+2x^{-1}+2)=440(x^2+x^{-2}+2x+2x^{-1}+3).$ Rearranging and letting $u=x+x^{-1} \Rightarrow u^2 - 2 = x^2 + x^{-2}$ gives us $u^2+2u-440=0.$ We take the positive root, so $u=20$ , which means $AC+CB=21(2+x+x^{-1})=21(2+u)=\boxed{462}$ .

Messy Trig Solution

Let $\theta$ be the smaller angle in the triangle. Then the sum of shorter and longer leg is $\sqrt{441}(2+\tan{\theta}+\cot{\theta})$ . We observe that the short leg has length $\sqrt{441}(1+\tan{\theta}) = \sqrt{440}(\sec{\theta}+\sin{\theta})$ . Grouping and squaring, we get $\sqrt{\frac{440}{441}} = \frac{\sin{\theta}+\cos{\theta}}{1+\sin{\theta}\cos{\theta}}$ . Squaring and using the double angle identity for sine, we get, $110(\sin{2\theta})^2 + \sin{2\theta} - 1 = 0$ . Solving, we get $\sin{2\theta} = \frac{1}{10}$ . Now to find $\tan{\theta}$ , we find $\cos{2\theta}$ using the Pythagorean Identity, and then use the tangent double angle identity. Thus, $\tan{\theta} = 10-3\sqrt{11}$ . Substituting into the original sum, we get $\boxed{462}$ .

Solution 4 (Algebra)

$[asy] size(200); pair A, B, C, D, E, F; A = (0, 5); B = (12, 0); C = (0, 0); D = (0, 60/17); E = (60/17, 60/17); F = (60/17, 0); draw(A--B--C--cycle); draw(D--E--F); label("$A$",A,N); label("$B$",B,dir(0)); label("$C$",C,SW); label("$D$",D,W); label("$E$",E,NE); label("$F$",F,S); label("$S_1$",(30/17,30/17)); [/asy]$

$[asy] size(200); pair A, B, C, W, X, Y, Z; A = (0, 5); B = (12, 0); C = (0, 0); real m = 1.31004366812; real n = 3.1441048035; W = (0,m); X = (m,m+n); Y = (m+n,n); Z = (n,0); draw(A--B--C--cycle); draw(W--X--Y--Z--cycle); label("$A$",A,N); label("$B$",B,dir(0)); label("$C$",C,SW); label("$W$",W,dir(180)); label("$X$",X,NE); label("$Y$",Y,NE); label("$Z$",Z,S); label("$S_2$",(2.22707423581,2.22707423581)); [/asy]$

Label points as above. Let $x=AC$ , $y=BC$ , $s_1 = 21$ be the side length of $S_1$ , and $s_2 = \sqrt{440}$ be the side length of $S_2$ .

Since $\triangle ABC\sim\triangle AED$ , we have $\frac{x}{y} = \frac{x-s_1}{s_1}$

$\implies xs_1=xy-ys_1$

$\implies xy=s_1(x+y)$

$\implies xy=21(x+y) \qquad \qquad (*)$ .

Since $\triangle ABC\sim\triangle AWX\sim\triangle ZBY$ , we have $s_2 + \frac{s_2x}{y} + \frac{s_2y}{x}=\sqrt{x^2+y^2}$

$\implies s_2(x^2+xy+y^2)=xy\sqrt{x^2+y^2}$

$\implies s_2^2(x^2+xy+y^2)^2 = x^2y^2(x^2+y^2)$

$\implies 440(x^2+xy+y^2)^2 = x^2y^2(x^2+y^2)$

Let $t=x+y$ . Repeatedly applying $(*)$ , we get $440(t^2-21t)^2 = 441t^2(t^2 - 42t)$ $440(t-21)^2 = 441(t^2-42t)$ $440t^2 - 42\cdot 440t + 440\cdot 441 = 441t^2 - 441\cdot 42t$ $t^2-42t-440\cdot 441=0$ $(t-21)^2 = 441^2$ $t-21=441$ $t=\boxed{462}$

~rayfish

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Last Question
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