To determine the structure of the quadrilateral formed by the points (6,8), (3,7), (-2,-2), and (1,-1), we will follow these steps:
### Step 1: Assign Points
Let's label the points:
- A(6, 8)
- B(3, 7)
- C(-2, -2)
- D(1, -1)
### Step 2: Calculate the Slopes
We will calculate the slopes of the sides AB, CD, AD, and BC using the slope formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
#### Slope of AB
Using points A(6, 8) and B(3, 7):
\[
\text{slope of AB} = \frac{7 - 8}{3 - 6} = \frac{-1}{-3} = \frac{1}{3}
\]
#### Slope of CD
Using points C(-2, -2) and D(1, -1):
\[
\text{slope of CD} = \frac{-1 - (-2)}{1 - (-2)} = \frac{1}{3}
\]
#### Slope of AD
Using points A(6, 8) and D(1, -1):
\[
\text{slope of AD} = \frac{-1 - 8}{1 - 6} = \frac{-9}{-5} = \frac{9}{5}
\]
#### Slope of BC
Using points B(3, 7) and C(-2, -2):
\[
\text{slope of BC} = \frac{-2 - 7}{-2 - 3} = \frac{-9}{-5} = \frac{9}{5}
\]
### Step 3: Analyze the Slopes
- The slopes of AB and CD are equal: \(\frac{1}{3}\), indicating that lines AB and CD are parallel.
- The slopes of AD and BC are equal: \(\frac{9}{5}\), indicating that lines AD and BC are also parallel.
### Step 4: Determine the Structure
Since both pairs of opposite sides are parallel, we can conclude that the quadrilateral is a **parallelogram**.
### Step 5: Check for Special Types
To determine if it is a rectangle, rhombus, or square, we need to check the lengths of the sides and the angles.
#### Length of AB
Using the distance formula:
\[
AB = \sqrt{(6-3)^2 + (8-7)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]
#### Length of CD
Using the distance formula:
\[
CD = \sqrt{(1 - (-2))^2 + (-1 - (-2))^2} = \sqrt{(1 + 2)^2 + (-1 + 2)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]
#### Length of AD
Using the distance formula:
\[
AD = \sqrt{(6-1)^2 + (8-(-1))^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106}
\]
#### Length of BC
Using the distance formula:
\[
BC = \sqrt{(3 - (-2))^2 + (7 - (-2))^2} = \sqrt{(3 + 2)^2 + (7 + 2)^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106}
\]
### Conclusion
Since the opposite sides are equal and parallel, but the lengths of adjacent sides are not equal, the quadrilateral is a **parallelogram** but not a rectangle, rhombus, or square.
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