A lot with two pairs of adjacent sides measuring 15m and 20m is to be fenced for subsequent landscaping works. The angle between the 15m and 20m sides is 79.60° and the angle between the longer sides is 80.94 degrees.

Find the following:
a. Length of fence needed
b. Length of dividing line to partition the lot into two equal areas
c. Area in sqm
d. If this area is to be covered by grass, how many 120cm by 20cm pieces of grass strips should the owner get?

To solve this problem, we need to analyze the geometry of the lot. Based on the given information, the lot can be modeled as a quadrilateral with known sides and angles. Here's how we can break down the problem:

### Given Data:
- Two adjacent sides of the lot: \( a = 15 \, \text{m} \) and \( b = 20 \, \text{m} \).
- The angle between these two sides: \( \alpha = 79.60^\circ \).
- The angle between the other two sides (longer sides): \( \beta = 80.94^\circ \).

### Step-by-Step Solution:

#### a. **Length of fence needed**
The total length of the fence is the perimeter of the quadrilateral. However, two sides are still unknown. We need to find the remaining two sides of the quadrilateral using the law of cosines.

For now, let’s call the unknown side lengths \( c \) and \( d \). These correspond to the sides opposite the two given angles.

We can calculate the missing sides using the known angles and dimensions of the lot. First, let’s start with the perimeter of the lot, which is the sum of the four sides.

#### b. **Length of dividing line (diagonal)**
The dividing line would partition the lot into two equal areas. Assuming that this line runs from the vertex of the 15 m and 20 m sides to the opposite corner, it forms two triangles with the known dimensions. We will calculate the diagonal length using geometric methods.

#### c. **Area in square meters**
The area can be found using the following formula for a quadrilateral:
\[
\text{Area} = \frac{1}{2}ab \sin(\alpha)
\]
This formula is derived from the formula for the area of a triangle, and since we have a quadrilateral composed of two triangles, we can apply it here.

#### d. **Number of grass strips needed**
We first need to compute the total area of the lot in square meters. Then, to find how many pieces of grass strips are required, we calculate the area of one strip (since each strip measures \( 120 \, \text{cm} \times 20 \, \text{cm} \), we can convert this to meters: \( 1.2 \, \text{m} \times 0.2 \, \text{m} \)) and divide the total area by the area of one strip.

Let's calculate each part in turn. I'll start with the calculations.### Calculations:

#### a. **Length of fence needed**:
We'll calculate this once we determine all sides of the lot using trigonometry.

#### c. **Area in square meters**:
The area of the lot is approximately:
\[
\text{Area} = 147.54 \, \text{sqm}
\]

#### d. **Number of grass strips needed**:
The area of one grass strip is \( 1.2 \, \text{m} \times 0.2 \, \text{m} = 0.24 \, \text{sqm} \). The number of strips required is approximately:
\[
614.73 \, \text{strips}
\]
So, the owner should get at least **615 pieces of grass strips**.

Next, I’ll calculate the length of the fence and dividing line.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the steps to complete the calculations manually. 

Would you like assistance with that, or any clarifications on the previous answers?

Here are five related questions to extend this topic:
1. How would the calculations change if one of the angles was a right angle?
2. How does the law of cosines help in solving unknown sides in a triangle?
3. What are the differences between various types of quadrilaterals when calculating area?
4. Could a different arrangement of the grass strips minimize wastage?
5. How can you find the perimeter of irregular polygons?

**Tip**: For angles in non-right triangles, the law of cosines is a powerful tool to find unknown sides or angles.

A lot with two pairs of adjacent sides measuring 15m and 20m is to be fenced for subsequent landscaping works. The angle between the 15m and 20m sides is 79.60° and the angle between the longer sides is 80.94 degrees. Find the following: a. Length of fence needed b. Length of dividing line to partition the lot into two equal areas c. Area in sqm d. If this area is to be covered by grass, how many 120cm by 20cm pieces of grass strips should the owner get?

Solve a problem involving a lot with sides of 15m and 20m, and angles of 79.60° and 80.94°. Learn how to calculate the length of the fence, the dividing diagonal, the area in square meters, and determine the number of grass strips needed for coverage. This problem covers geometry, trigonometry, and area calculations, ideal for Grades 9-12 students.

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