A rectangle has length $40\,\text{cm}$ and width $16\,\text{cm}$. Point $M$ is the midpoint of one side. The paper is folded along a line through $M$. If a vertex of the side containing $M$ does not land on the opposite side after folding, find the length of the crease (in cm). Fill in the blank: $\underline{\quad\quad}$.

Step 1: Define Coordinates and Points

Let's set up a coordinate system for the rectangle. We place vertex $D$ at the origin $(0{,}0)$. The length of the rectangle is $40\,\text{cm}$ and the width is $16\,\text{cm}$. So the vertices are $A(0{,}16)$, $B(40{,}16)$, $C(40{,}0)$, and $D(0{,}0)$. Point $M$ is the midpoint of side $CD$, so its coordinates are $(20{,}0)$.

Step 2: Determine Folded Vertex Position

When the paper is folded along a line through $M$, vertex $D$ moves to a new position, let's call it $D'$. The problem states that $D'$ does not land on the opposite side $AB$. This means $D'$ must lie on the side $BC$ or $AD$, or outside the rectangle. Since $M$ is on $CD$, and we are folding a vertex from $CD$, $D'$ will be on $BC$ or $AD$. Given the setup, $D'$ will land on side $BC$. Let $D'=(x,y)$.

Step 3: Apply Properties of Folding

Folding implies that the distance from $M$ to $D$ is equal to the distance from $M$ to $D'$. Since $M=(20{,}0)$ and $D=(0{,}0)$, $MD = 20\,\text{cm}$. Therefore, $MD' = 20\,\text{cm}$. Since $D'$ lies on side $BC$, its $x$ -coordinate must be $40$. So $D'=(40, y)$.

Step 4: Calculate $y$ -coordinate of $D'$

Using the distance formula for $MD'$, we have $(40-20)^2 + (y-0)^2 = 20^2$. This simplifies to $20^2 + y^2 = 20^2$, which means $y^2 = 0$, so $y=0$. This implies $D'$ is at $(40{,}0)$, which is point $C$.

Step 5: Determine the Crease Line

The crease line is the perpendicular bisector of the segment connecting the original point $D$ and its folded position $D'$. Since $D=(0{,}0)$ and $D'=(40{,}0)$, the segment $DD'$ lies on the $x$ -axis. The midpoint of $DD'$ is $(20{,}0)$, which is $M$. The perpendicular bisector of $DD'$ is a vertical line $x=20$. However, the crease must pass through $M$. This means the crease is the line segment from $M$ to a point on the top side $AB$. Let the crease be $MN$, where $N$ is on $AB$.

Step 6: Find the Length of the Crease

Since the crease is the line segment $MN$ where $M=(20{,}0)$ and $N$ is on $AB$ with $x$ -coordinate $20$, $N=(20{,}16)$. The length of the crease $MN$ is the distance between $(20{,}0)$ and $(20{,}16)$, which is $16-0 = 16\,\text{cm}$.

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