In a farmer's quest to plant hibiscus flowers, the ratio of one square unit area of land to the number of hibiscus flowers is 1:5. the land for the farm is in the shape of a trapezium with vertices ABCD. the coordinates of three of the vertices are A(11, 4), B(7,7), and C(-3, 2). The side AD is parallel to BC and the side CD is perpendicular to BC. Find the number of hibiscus flowers that the farmer can plant on his farm.

To solve this problem, we need to determine the area of the trapezium-shaped land and then use the given ratio to find the number of hibiscus flowers that can be planted.

### Step 1: Determine the coordinates of point D
Since AD is parallel to BC, and CD is perpendicular to BC, we need to find the coordinates of point D.

1. **Find the slope of BC**:
   - Coordinates of B: \((7, 7)\)
   - Coordinates of C: \((-3, 2)\)
   - Slope of \( BC \) is:
     \[
     \text{slope of } BC = \frac{7 - 2}{7 - (-3)} = \frac{5}{10} = \frac{1}{2}
     \]

2. **Since AD is parallel to BC**, the slope of \( AD \) will also be \( \frac{1}{2} \).

3. **Since CD is perpendicular to BC**, the slope of \( CD \) will be the negative reciprocal of \( \frac{1}{2} \), which is \( -2 \).

### Step 2: Set up the equation for line CD
The equation of line \( CD \) can be written using the point-slope form:
\[
y - y_C = m \cdot (x - x_C)
\]
Substitute \( m = -2 \), and point \( C(-3, 2) \):
\[
y - 2 = -2(x + 3)
\]
Simplify:
\[
y = -2x - 6 + 2
\]
\[
y = -2x - 4
\]

### Step 3: Find the intersection of AD with CD to determine D
Let \( D(x_D, y_D) \) be the point on line \( CD \) where \( AD \) intersects. The equation of line \( AD \) is:
\[
y - y_A = \frac{1}{2}(x - x_A)
\]
Substitute point \( A(11, 4) \):
\[
y - 4 = \frac{1}{2}(x - 11)
\]
Simplify:
\[
y = \frac{1}{2}x - \frac{11}{2} + 4 = \frac{1}{2}x - \frac{3}{2}
\]

Since point \( D \) lies on both lines:
\[
\frac{1}{2}x_D - \frac{3}{2} = -2x_D - 4
\]
Multiply by 2 to clear the fraction:
\[
x_D - 3 = -4x_D - 8
\]
Combine like terms:
\[
5x_D = -5
\]
\[
x_D = -1
\]

Substitute \( x_D = -1 \) into the equation for \( CD \) to find \( y_D \):
\[
y_D = -2(-1) - 4 = 2 - 4 = -2
\]

Thus, point \( D \) is \((-1, -2)\).

### Step 4: Calculate the area of the trapezium
The length of the parallel sides \( AD \) and \( BC \) and the height \( CD \) are needed to calculate the area.

1. **Length of BC**:
\[
\text{Length of } BC = \sqrt{(7 - (-3))^2 + (7 - 2)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5}
\]

2. **Length of AD**:
\[
\text{Length of } AD = \sqrt{(11 - (-1))^2 + (4 - (-2))^2} = \sqrt{12^2 + 6^2} = \sqrt{144 + 36} = \sqrt{180} = 6\sqrt{5}
\]

3. **Height \( CD \)**:
Since \( D(-1, -2) \) and \( C(-3, 2) \), we have:
\[
\text{Length of } CD = \sqrt{(-1 - (-3))^2 + (-2 - 2)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

4. **Area of the trapezium**:
\[
\text{Area} = \frac{1}{2} \times (\text{Length of } AD + \text{Length of } BC) \times \text{Height}
\]
\[
\text{Area} = \frac{1}{2} \times (6\sqrt{5} + 5\sqrt{5}) \times 2\sqrt{5}
\]
\[
\text{Area} = \frac{1}{2} \times 11\sqrt{5} \times 2\sqrt{5} = \frac{1}{2} \times 11 \times 10 = 55 \text{ square units}
\]

### Step 5: Calculate the number of hibiscus flowers
Given the ratio of area to the number of hibiscus flowers is 1:5:
\[
\text{Number of hibiscus flowers} = 55 \times 5 = 275
\]

Thus, the farmer can plant **275 hibiscus flowers** on his farm.

Would you like more details or have any questions? Here are some related questions:

1. What are the properties of a trapezium and how are they used in calculations?
2. How do you find the length of a line segment given two points in a coordinate plane?
3. Can the equation of a line be derived in different forms, and when should each be used?
4. What are some applications of finding the area of irregular shapes in real-life situations?
5. How do slopes of parallel and perpendicular lines relate to each other?

**Tip:** When working with geometric shapes on a coordinate plane, drawing a rough sketch often helps visualize and solve the problem more efficiently.

Explore how to determine the number of hibiscus flowers a farmer can plant on a trapezium-shaped land. Step-by-step solution includes finding coordinates, calculating area, and applying ratios. Suitable for Grades 10-12 students and anyone interested in coordinate geometry.

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