To solve this problem, we need to visualize and analyze the given geometric configuration step by step.

Step 1: Understand the Geometry and Given Information

• Trapezoid $ABCD$: $AD$ is parallel to $BC$.
• $BD$ is perpendicular to $DC$: This implies $\triangle BDC$ is a right triangle.
• $F$ is on $BD$ such that $AF$ is perpendicular to $BD$.
• $AF$ is extended to meet $BC$ at $E$.
• Given lengths: $AB = 41$, $AD = 50$, $BF = 9$.

Step 2: Assign Coordinates

Let's place the trapezoid on the coordinate plane:

1. Point $A$ at $(0, 0)$ (since $AD$ is horizontal, start with $A$).
2. Point $D$ at $(50, 0)$ (since $AD = 50$ and $AD$ is parallel to $x$-axis).
3. $BD$ perpendicular to $DC$ implies a vertical line $BD$ and a horizontal line $DC$.
4. Place $B$ on $BD$ and $C$ on $BC$. Assume $D$ is below $B$ vertically because $BD$ is perpendicular to $DC$.

Step 3: Coordinate Assignments

• Let $B$ be at $(50, b)$ for some $b > 0$.
• Since $BD$ is vertical, $D$ is at $(50, 0)$.
• $F$ is on $BD$, and $BF = 9$. Thus, $F$ is at $(50, b-9)$.
• $C$ is on $BC$, directly horizontal from $D$. Let $C$ be at $(c, 0)$.

Step 4: Calculate $b$

Since $AB = 41$ and $AF$ is perpendicular to $BD$, point $A$ and $B$ form a right triangle with $F$ as a leg:

• $AB^2 = BF^2 + AF^2$.
• $41^2 = 9^2 + AF^2$.
• $1681 = 81 + AF^2$.
• $AF^2 = 1600$.
• $AF = 40$.

Thus, $F$ is $40$ units horizontally from $A$:

• $F = (0, b - 9)$ implies $b - 9 = 40$.
• $b = 49$.

So, $B$ is at $(50, 49)$ and $F$ is at $(50, 40)$.

Step 5: Find Coordinates of $E$

Since $AF$ is extended to $E$ and $E$ lies on $BC$, we know:

• $AF$ has the slope $\frac{49-0}{50-0} = \frac{49}{50}$.
• Equation of line $AF$: $y = \frac{49}{50}x$.
• $E$ is on line $BC$, with $BC$ having slope $0$ (horizontal line).
• Hence, $E$ lies on $BC$: $y = 49$.

Find $x$ where $y = 49$:

• $49 = \frac{49}{50}x$.
• $x = 50$.

Thus, $E$ is at $(50, 49)$.

Step 6: Coordinates of $C$ and Area of Quadrilateral $FECD$

From the previous setup:

• $C$ is horizontal from $D$: $C = (c, 0)$, assume $c = 50$ for simplicity.

Step 7: Calculate Area of Quadrilateral $FECD$

1. $F = (50, 40)$.
2. $E = (50, 49)$.
3. $C = (50, 0)$.
4. $D = (0, 0)$.

Area of Quadrilateral $FECD$ is a combination of a rectangle $FECD$:

• $FE = 49 - 40 = 9$ (height).
• $DC = 50$ (width).

The area is given by:

$\text{Area} = 9 \times 50 = 450$

Therefore, the area of quadrilateral $FECD$ is 450 square units.

Would you like more details, or do you have any questions?

Related Questions

1. What is the perimeter of the trapezoid $ABCD$?
2. How can we find the coordinates of $C$ using another method?
3. What is the distance between points $A$ and $F$?
4. Can the area of $FECD$ be found using the shoelace formula?
5. How would the solution change if $AB$ was not perpendicular to $BD$?

Tip

When solving geometry problems, always start by defining the coordinates for key points based on given conditions to simplify calculations.