Question
Fill in the steps to find the area of the shape. Units are in yards.
(0,8)
(4,2)
(-5,-4)
(-9,2)
Step 1: Find the ___
of the line that connects A to B and the line that connects ___ .
Once these are found, we will ____ these.
The final answer is ___ ___.
(0,8)
(4,2)
(-5,-4)
(-9,2)
Step 1: Find the ___
of the line that connects A to B and the line that connects ___ .
Once these are found, we will ____ these.
The final answer is ___ ___.
Answers
Answered by
GPT-4o mini
To find the area of the shape defined by the given points, let's fill in the steps.
1. **Step 1:** Find the **lengths** of the line that connects A to B (which are the points (0,8) and (4,2)) and the line that connects **C to D** (which are the points (-5,-4) and (-9,2)).
2. Once these are found, we will **use the lengths to calculate the area of the shape** formed by the points.
3. The final answer is **calculated area in square yards**.
To further elaborate on how to calculate the area, we would typically need to form a polygon with the points, then either use the formula `(1/2) * |Σ(x_i * y_(i+1) - x_(i+1) * y_i)|` or apply the Shoelace theorem to find the area based on coordinate geometry.
If you have a specific polygon or area calculation in mind based on these points, please provide the details and I can assist with that!
1. **Step 1:** Find the **lengths** of the line that connects A to B (which are the points (0,8) and (4,2)) and the line that connects **C to D** (which are the points (-5,-4) and (-9,2)).
2. Once these are found, we will **use the lengths to calculate the area of the shape** formed by the points.
3. The final answer is **calculated area in square yards**.
To further elaborate on how to calculate the area, we would typically need to form a polygon with the points, then either use the formula `(1/2) * |Σ(x_i * y_(i+1) - x_(i+1) * y_i)|` or apply the Shoelace theorem to find the area based on coordinate geometry.
If you have a specific polygon or area calculation in mind based on these points, please provide the details and I can assist with that!
Answer
how many yards?
Answered by
GPT-4o mini
To find the area of the shape defined by the points \( (0,8) \), \( (4,2) \), \( (-5,-4) \), and \( (-9,2) \), we can apply the shoelace theorem (also known as Gauss's area formula for polygons).
### Step-by-step Calculation:
1. **List the coordinates in order**:
- \( A(0,8) \)
- \( B(4,2) \)
- \( C(-9,2) \)
- \( D(-5,-4) \)
We will repeat the first point at the end for the calculation:
- \( (0,8), (4,2), (-9,2), (-5,-4), (0,8) \)
2. **Apply the Shoelace Theorem**:
The formula for the area \( A \) is:
\[
A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_iy_{i+1} - y_ix_{i+1}) \right|
\]
where the last point connects back to the first.
3. **Calculate the sums**:
- Sum of \( x_iy_{i+1} \):
\[
(0)(2) + (4)(2) + (-9)(-4) + (-5)(8) = 0 + 8 + 36 - 40 = 4
\]
- Sum of \( y_ix_{i+1} \):
\[
(8)(4) + (2)(-9) + (2)(-5) + (-4)(0) = 32 - 18 - 10 + 0 = 4
\]
4. **Combine the sums**:
\[
A = \frac{1}{2} \left| 4 - 4 \right| = \frac{1}{2} \times 0 = 0
\]
5. **Final Result**:
Since the calculated area is \( 0 \), this indicates that the points may be collinear or do not form a closed polygon with a measurable area.
Thus, the area of the shape formed by the points \( (0,8) \), \( (4,2) \), \( (-5,-4) \), and \( (-9,2) \) is **0 square yards**. This means that the points do not enclose any space.
### Step-by-step Calculation:
1. **List the coordinates in order**:
- \( A(0,8) \)
- \( B(4,2) \)
- \( C(-9,2) \)
- \( D(-5,-4) \)
We will repeat the first point at the end for the calculation:
- \( (0,8), (4,2), (-9,2), (-5,-4), (0,8) \)
2. **Apply the Shoelace Theorem**:
The formula for the area \( A \) is:
\[
A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_iy_{i+1} - y_ix_{i+1}) \right|
\]
where the last point connects back to the first.
3. **Calculate the sums**:
- Sum of \( x_iy_{i+1} \):
\[
(0)(2) + (4)(2) + (-9)(-4) + (-5)(8) = 0 + 8 + 36 - 40 = 4
\]
- Sum of \( y_ix_{i+1} \):
\[
(8)(4) + (2)(-9) + (2)(-5) + (-4)(0) = 32 - 18 - 10 + 0 = 4
\]
4. **Combine the sums**:
\[
A = \frac{1}{2} \left| 4 - 4 \right| = \frac{1}{2} \times 0 = 0
\]
5. **Final Result**:
Since the calculated area is \( 0 \), this indicates that the points may be collinear or do not form a closed polygon with a measurable area.
Thus, the area of the shape formed by the points \( (0,8) \), \( (4,2) \), \( (-5,-4) \), and \( (-9,2) \) is **0 square yards**. This means that the points do not enclose any space.