Asked by ASDF

The bases of trapezoid ABCD are AB and CD. Let P be the intersection of diagonals AC and BD. If the areas of triangles ABP and CDP are 8 and 18, respectively, then find the area of trapezoid ABCD.
Answered by Anonymous
50
Answered by ANYONUMES
Triangles $PAB$ and $PCD$ are similar, so
\[\frac{[PAB]}{[PCD]} = \left( \frac{AB}{CD} \right)^2 = \frac{x^2}{y^2}.\]
But we are given that $[PAB] = 8$ and $[PCD] = 18$, so
\[\frac{x^2}{y^2} = \frac{8}{18} = \frac{4}{9},\]
which means $x/y = 2/3$.

Then $AP/PC = AB/CD = 2/3$. Triangles $ABP$ and $BCP$ have the same height with respect to base $\overline{AC}$, so
\[\frac{[BCP]}{[ABP]} = \frac{CP}{AP} = \frac{3}{2},\]
which means $[BCP] = 3/2 \cdot [ABP] = 3/2 \cdot 8 = 12$.

Also, $BP/PD = AB/CD = 2/3$. Triangles $ABP$ and $ADP$ have the same height with respect to base $\overline{BD}$, so
\[\frac{[ADP]}{[ABP]} = \frac{DP}{BP} = \frac{3}{2},\]
which means $[ADP] = 3/2 \cdot [ABP] = 3/2 \cdot 8 = 12$.

Therefore, the area of trapezoid $ABCD$ is $[ABP] + [BCP] + [CDP] + [DAP] = 8 + 12 + 18 + 12 = \boxed{50}$.
Answered by asdfjkl;
Thanks for copying and pasting LaTeX and not bothering to put it so its legible.
Answered by AoPS
Hello,

Our challenge problems are designed for you to solve them to the best of your ability. If you need any guidance, ask on our message boards.

Sincerely,
AoPS
Answered by AoPS
Actually, the person above was not the real AoPS. Our message board sometimes have problems and we encourage you to post your answers here
Answered by Anonymous
@above dont lie you know you are not supposed post AoPS answers here
Answered by Anonymous
@fake aops it is "Our message board sometimes has problems" not "Our message board sometimes have problems"
Answered by AoPS
lmao u guys should quit with the fake aops...
you guys are wasting your money and your life for looking for answers online.
not that I care but you should... whatevs.
Answered by Anonymous
Well said
Answered by Anonymous
There was an honor code that everyone had to agree with, so posting AoPS's answers on here is "double-violating" the code
Answered by Anonymous
yeah dont do that
Answered by AoPS
Do not cheat you are basically very wasteful
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