We are tasked to determine the length of $CF$ in a rectangle $ABCD$, where $E$ is the midpoint of $BC$, $F$ is the midpoint of $CD$, and $\angle AFE = 90^\circ$. Additionally, we know that $BC = 2$. Let us solve this step-by-step.

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Step 1: Assign coordinates to the vertices of the rectangle

Let $A = (0, 0)$, $B = (2a, 0)$, $C = (2a, 2)$, and $D = (0, 2)$, where $2a$ is the length of the rectangle and $2$ is the height (since $BC = 2$).

Midpoints:

• $E$, the midpoint of $BC$, is at: $E = \left(\frac{2a + 2a}{2}, \frac{0 + 2}{2}\right) = (2a, 1)$
• $F$, the midpoint of $CD$, is at: $F = \left(\frac{2a + 0}{2}, \frac{2 + 2}{2}\right) = (a, 2)$

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Step 2: Analyze the given angle $\angle AFE = 90^\circ$

For $\angle AFE = 90^\circ$, the vectors $\overrightarrow{AF}$ and $\overrightarrow{EF}$ must be perpendicular. This means their dot product is zero: $\overrightarrow{AF} \cdot \overrightarrow{EF} = 0$

Compute $\overrightarrow{AF}$:

$\overrightarrow{AF} = (a, 2) - (0, 0) = (a, 2)$

Compute $\overrightarrow{EF}$:

$\overrightarrow{EF} = (a, 2) - (2a, 1) = (-a, 1)$

Compute the dot product:

$\overrightarrow{AF} \cdot \overrightarrow{EF} = (a)(-a) + (2)(1) = -a^2 + 2$

For perpendicularity, set the dot product to zero: $-a^2 + 2 = 0$ $a^2 = 2 \quad \Rightarrow \quad a = \sqrt{2}$

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Step 3: Calculate $CF$

From the coordinates of $C$ and $F$: $C = (2a, 2) = (2\sqrt{2}, 2), \quad F = (a, 2) = (\sqrt{2}, 2)$

The length $CF$ is: $CF = \sqrt{(2\sqrt{2} - \sqrt{2})^2 + (2 - 2)^2}$ $CF = \sqrt{(\sqrt{2})^2} = \sqrt{2}$

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Final Answer:

$CF = \sqrt{2}$

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Would you like detailed clarifications on any part of this solution? Let me know!

Related Questions:

1. How do you find the midpoint of a line segment given endpoints?
2. Can you explain how dot products determine perpendicularity?
3. What is the geometric significance of a 90-degree angle in vector analysis?
4. How would the solution change if $BC \neq 2$?
5. Can you explain the role of coordinate geometry in solving similar problems?

Tip:

When analyzing geometric problems, assigning coordinates simplifies calculations and helps verify conditions like perpendicularity or collinearity easily.