To solve the problem step by step, we will follow these instructions:
### Step 1: Plot the Points A, B, C, and D
We are given the coordinates of the points:
- A(1, 1)
- B(5, 1)
- C(4, 2)
- D(2, 2)
On a graph, plot these points:
- Point A is at (1, 1)
- Point B is at (5, 1)
- Point C is at (4, 2)
- Point D is at (2, 2)
### Step 2: Connect the Points to Form Quadrilateral ABCD
Once the points are plotted, connect them in the order A, B, C, D to form the quadrilateral ABCD.
### Step 3: Reflect Points A, B, C, and D in the Origin
To reflect a point (x, y) in the origin, the new coordinates will be (-x, -y). Therefore, we calculate the reflected points:
- A' = (-1, -1) (from A(1, 1))
- B' = (-5, -1) (from B(5, 1))
- C' = (-4, -2) (from C(4, 2))
- D' = (-2, -2) (from D(2, 2))
### Step 4: Plot the Reflected Points A', B', C', and D'
Now, plot the reflected points on the graph:
- A' is at (-1, -1)
- B' is at (-5, -1)
- C' is at (-4, -2)
- D' is at (-2, -2)
### Step 5: Connect the Reflected Points
Connect the points A', B', C', and D' to visualize the reflected quadrilateral.
### Step 6: Check Collinearity of Points D, A, A', and D'
To check if points D, A, A', and D' are collinear, we can use the slope formula. If the slope between any two pairs of points is the same, then the points are collinear.
1. Calculate the slope between D(2, 2) and A(1, 1):
\[
\text{slope}_{DA} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{1 - 2} = \frac{-1}{-1} = 1
\]
2. Calculate the slope between A(1, 1) and A'(-1, -1):
\[
\text{slope}_{AA'} = \frac{-1 - 1}{-1 - 1} = \frac{-2}{-2} = 1
\]
3. Calculate the slope between A'(-1, -1) and D'(-2, -2):
\[
\text{slope}_{A'D'} = \frac{-2 - (-1)}{-2 - (-1)} = \frac{-1}{-1} = 1
\]
Since all slopes are equal (1), points D, A, A', and D' are collinear.
### Final Answer
The coordinates of the reflected points are:
- A'(-1, -1)
- B'(-5, -1)
- C'(-4, -2)
- D'(-2, -2)
Yes, points D, A, A', and D' are collinear.
