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Difference between revisions of "2018 AIME II Problems/Problem 12"

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<math>\frac{h_1}{2}AP + \frac{h_2}{2}CP = \frac{h_1}{2}CP + \frac{h_2}{2}AP \rightarrow \left(h_1 - h_2\right)AP = \left(h_1 - h_2\right)CP \rightarrow AP = CP</math>.
 
<math>\frac{h_1}{2}AP + \frac{h_2}{2}CP = \frac{h_1}{2}CP + \frac{h_2}{2}AP \rightarrow \left(h_1 - h_2\right)AP = \left(h_1 - h_2\right)CP \rightarrow AP = CP</math>.
  
−
What an important finding! Note that the opposite sides <math>\overline{AB}</math> and <math>\overline{CD}</math> have equal length, and note that diagonal <math>\overline{DB}</math> bisects diagonal <math>\overline{AC}</math>. This is very similar to what happens if <math>ABCD</math> were a parallelogram with <math>AB = CD = 10</math>, so let's extend <math>\overline{DB}</math> to point <math>E</math>, such that <math>AECD</math> is a parallelogram. In other words, <math>AE = CD = 10</math> and <math>EC = DA = 2\sqrt{65}</math>. Now, let's examine <math>\triangle ABE</math>. Since <math>AB = AE = 10</math>, the triangle is isosceles, and <math>\angle ABE \cong \angle AEB</math>. Note that in parallelogram <math>AECD</math>, <math>\angle AED</math> and <math>\angle CDE</math> are congruent, so <math>\angle ABE \cong \angle CDE</math> and thus <math>\text{m}\angle ABD = 180^\circ - \text{m}\angle CDB</math>. Define <math>\alpha := \text{m}\angle CDB</math>, so <math>180^\circ - \alpha = \text{m}\angle ABD</math>. We use the Law of Cosines on <math>\triangle DAB</math> and <math>\triangle CAB</math>:
+
What an important finding! Note that the opposite sides <math>\overline{AB}</math> and <math>\overline{CD}</math> have equal length, and note that diagonal <math>\overline{DB}</math> bisects diagonal <math>\overline{AC}</math>. This is very similar to what happens if <math>ABCD</math> were a parallelogram with <math>AB = CD = 10</math>, so let's extend <math>\overline{DB}</math> to point <math>E</math>, such that <math>AECD</math> is a parallelogram. In other words, <math>AE = CD = 10</math> and <math>EC = DA = 2\sqrt{65}</math>. Now, let's examine <math>\triangle ABE</math>. Since <math>AB = AE = 10</math>, the triangle is isosceles, and <math>\angle ABE \cong \angle AEB</math>. Note that in parallelogram <math>AECD</math>, <math>\angle AED</math> and <math>\angle CDE</math> are congruent, so <math>\angle ABE \cong \angle CDE</math> and thus <math>\text{m}\angle ABD = 180^\circ - \text{m}\angle CDB</math>. Define <math>\alpha := \text{m}\angle CDB</math>, so <math>180^\circ - \alpha = \text{m}\angle ABD</math>. We use the Law of Cosines on <math>\triangle DAB</math> and <math>\triangle CDB</math>:
  
 
<math>\left(2\sqrt{65}\right)^2 = 10^2 + BD^2 - 20BD\cos\left(180^\circ - \alpha\right) = 100 + BD^2 + 20BD\cos\alpha,</math>
 
<math>\left(2\sqrt{65}\right)^2 = 10^2 + BD^2 - 20BD\cos\left(180^\circ - \alpha\right) = 100 + BD^2 + 20BD\cos\alpha,</math>

Revision as of 06:36, 31 March 2018

Problem

Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.

Solution

For reference, $2\sqrt{65} \approx 16$, so $\overline{AD}$ is the longest of the four sides of $ABCD$. Let $h_1$ be the length of the altitude from $B$ to $\overline{AC}$, and let $h_2$ be the length of the altitude from $D$ to $\overline{AC}$. Then, the triangle area equation becomes

$\frac{h_1}{2}AP + \frac{h_2}{2}CP = \frac{h_1}{2}CP + \frac{h_2}{2}AP \rightarrow \left(h_1 - h_2\right)AP = \left(h_1 - h_2\right)CP \rightarrow AP = CP$.

What an important finding! Note that the opposite sides $\overline{AB}$ and $\overline{CD}$ have equal length, and note that diagonal $\overline{DB}$ bisects diagonal $\overline{AC}$. This is very similar to what happens if $ABCD$ were a parallelogram with $AB = CD = 10$, so let's extend $\overline{DB}$ to point $E$, such that $AECD$ is a parallelogram. In other words, $AE = CD = 10$ and $EC = DA = 2\sqrt{65}$. Now, let's examine $\triangle ABE$. Since $AB = AE = 10$, the triangle is isosceles, and $\angle ABE \cong \angle AEB$. Note that in parallelogram $AECD$, $\angle AED$ and $\angle CDE$ are congruent, so $\angle ABE \cong \angle CDE$ and thus $\text{m}\angle ABD = 180^\circ - \text{m}\angle CDB$. Define $\alpha := \text{m}\angle CDB$, so $180^\circ - \alpha = \text{m}\angle ABD$. We use the Law of Cosines on $\triangle DAB$ and $\triangle CDB$:

$\left(2\sqrt{65}\right)^2 = 10^2 + BD^2 - 20BD\cos\left(180^\circ - \alpha\right) = 100 + BD^2 + 20BD\cos\alpha,$

$14^2 = 10^2 + BD^2 - 20BD\cos\alpha.$

Subtracting the second equation from the first yields

$260 - 196 = 40BD\cos\alpha \rightarrow BD\cos\alpha = \frac{8}{5}.$

This means that dropping an altitude from $B$ to some foot $Q$ on $\overline{CD}$ gives $DQ = \frac{8}{5}$ and therefore $CQ = \frac{42}{5}$. Seeing that $CQ = \frac{3}{5}\cdot BC$, we conclude that $\triangle QCB$ is a 3-4-5 right triangle, so $BQ = \frac{56}{5}$. Then, the area of $\triangle BCD$ is $\frac{1}{2}\cdot 10 \cdot \frac{56}{5} = 56$. Since $AP = CP$, points $A$ and $C$ are equidistant from $\overline{BD}$, so $\left[\triangle ABD\right] = \left[\triangle CBD\right] = 56$ and hence $\left[ABCD\right] = 56 + 56 = \boxed{112}$. -kgator


2018 AIME II (Problems • Answer Key • Resources)
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