HMMT-Nov 2023 guts, Problem 3

Let $A B C D$ be a rectangle with $A B=20$ and $A D=23$. Let $M$ be the midpoint of $C D$, and let $X$ be the reflection of $M$ across point $A$. Compute the area of triangle $X B D$.

Solution: Solution: Observe that $[X B D]=[B A D]+[B A X]+[D A X]$. We will find the area of each of these triangles individually. \begin{itemize}   \item We have $[A B D]=\frac{1}{2}[A B C D]$.   \item Because $A M=A X,[B A X]=[B A M]$ as the triangles have the same base and height. Thus, as $[B A M]$ have the same base and height as $A B C D,[B A X]=[B A M]=\frac{1}{2}[A B C D]$.   \item From similar reasoning, we know that $[D A X]=[D A M]$. We have that $D A M$ has the same base and half the height of the rectangle. Thus, $[D A X]=[D A M]=\frac{1}{4}[A B C D]$. \end{itemize} Hence, we have \[ \begin{aligned} {[X B D] } & =[B A D]+[B A X]+[D A X] \\ & =\frac{1}{2}[A B C D]+\frac{1}{2}[A B C D]+\frac{1}{4}[A B C D] \\ & =\frac{5}{4}[A B C D] \end{aligned} \] Thus, our answer is $\frac{5}{4}[A B C D]=\frac{5}{4}(20 \cdot 23)=575$. $\fbox{575}$.

Answer: 575