In trapezoid ABCD, AD is parallel to BC. Also, BD is perpendicular to DC. The point F is chosen on line BD so that AF is perpendicular to BD. AF is extended to meet BC at point E. If AB = 41, AD = 50 and BF = 9, what is the area of quadrilateral FECD?

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.

Learn how to calculate the area of quadrilateral FECD in trapezoid ABCD, where AD is parallel to BC and BD is perpendicular to DC. Given dimensions AB = 41, AD = 50, and BF = 9, discover the geometric configuration and step-by-step solution. Suitable for students in Grades 9-12.

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