in the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, -1) respectively.
1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure?
2. calculate the area of the figure formed by points a, b, and c.
3. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

Question

Accepted Answer

### Sub - Question 1
# Explanation:
## Step 1: Analyze coordinates of points
Point $ A(1,2) $, $ B(4,2) $, $ C(4, - 1) $. The $ y $ - coordinate of $ A $ and $ B $ is the same ($ y = 2 $), so $ AB $ is a horizontal line segment. The $ x $ - coordinate of $ B $ and $ C $ is the same ($ x = 4 $), so $ BC $ is a vertical line segment.
## Step 2: Determine the angle between $ AB $ and $ BC $
Since $ AB $ is horizontal and $ BC $ is vertical, the angle between $ AB $ and $ BC $ is $ 90^{\circ} $. Also, we have three points, so the figure is a right - angled triangle.
# Answer:
The figure is a right - angled triangle.

### Sub - Question 2
# Explanation:
## Step 1: Calculate the length of $ AB $
For two points $ (x_1,y_1) $ and $ (x_2,y_2) $ with the same $ y $ - coordinate, the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $ simplifies to $ |x_2 - x_1| $. For $ A(1,2) $ and $ B(4,2) $, $ AB=\vert4 - 1\vert=3 $.
## Step 2: Calculate the length of $ BC $
For two points $ (x_1,y_1) $ and $ (x_2,y_2) $ with the same $ x $ - coordinate, the distance formula simplifies to $ |y_2 - y_1| $. For $ B(4,2) $ and $ C(4,-1) $, $ BC=\vert2-(-1)\vert = 3 $.
## Step 3: Calculate the area of the right - angled triangle
The area of a right - angled triangle is given by $ A=\frac{1}{2}\times\text{base}\times\text{height} $. Here, base $ AB = 3 $ and height $ BC = 3 $, so $ A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5 $.
# Answer:
The area of the figure is $ 4.5 $ (or $ \frac{9}{2} $).

### Sub - Question 3
# Explanation:
## Step 1: Recall the properties of a rectangle
In a rectangle $ ABCD $, $ AB\parallel CD $ and $ AD\parallel BC $, and $ AB = CD $, $ AD = BC $. Since $ AB $ is horizontal from $ (1,2) $ to $ (4,2) $, $ CD $ should also be horizontal. Since $ BC $ is vertical from $ (4,2) $ to $ (4,-1) $, $ AD $ should also be vertical.
## Step 2: Determine the coordinates of $ D $
The $ x $ - coordinate of $ D $ should be the same as the $ x $ - coordinate of $ A $ (because $ AD $ is vertical), so $ x = 1 $. The $ y $ - coordinate of $ D $ should be the same as the $ y $ - coordinate of $ C $ (because $ CD $ is horizontal), so $ y=-1 $. So the coordinates of $ D $ are $ (1,-1) $.
## Step 3: Verify the rectangle
- Check $ AB $ and $ CD $: $ AB $ has length $ 3 $ (from $ x = 1 $ to $ x = 4 $ at $ y = 2 $), $ CD $ has length $ 3 $ (from $ x = 1 $ to $ x = 4 $ at $ y=-1 $), and they are both horizontal, so $ AB\parallel CD $ and $ AB = CD $.
- Check $ BC $ and $ AD $: $ BC $ has length $ 3 $ (from $ y = 2 $ to $ y=-1 $ at $ x = 4 $), $ AD $ has length $ 3 $ (from $ y = 2 $ to $ y=-1 $ at $ x = 1 $), and they are both vertical, so $ BC\parallel AD $ and $ BC = AD $. Also, all angles are $ 90^{\circ} $ (since $ AB $ is horizontal and $ BC $ is vertical, $ AD $ is vertical and $ CD $ is horizontal).
# Answer:
The coordinates of point $ D $ are $ (1,-1) $.

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QUESTION IMAGE

Question

Explanation:

Sub - Question 1

Step 1: Analyze coordinates of points

Step 2: Determine the angle between \( AB \) and \( BC \)

Step 1: Calculate the length of \( AB \)

Step 2: Calculate the length of \( BC \)

Step 3: Calculate the area of the right - angled triangle

Step 1: Recall the properties of a rectangle

Step 2: Determine the coordinates of \( D \)

Step 3: Verify the rectangle

Answer:

Sub - Question 2