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Sagot :
To draw the rectangle on the coordinate plane, we can proceed step by step by first establishing the given coordinates and then finding the dimensions and other vertices of the rectangle.
### Step-by-Step Solution
1. Given Coordinates of the Upper Veritices:
- Upper-left corner is at [tex]\((-1, 7)\)[/tex].
- Upper-right corner is at [tex]\((4, 7)\)[/tex].
2. Calculate the Width:
- The width of the rectangle is the horizontal distance between the upper-left and upper-right vertices.
- The x-coordinate of the upper-left corner is [tex]\(-1\)[/tex].
- The x-coordinate of the upper-right corner is [tex]\(4\)[/tex].
- Width = [tex]\(4 - (-1) = 5\)[/tex] units.
3. Calculate the Height:
- The area of the rectangle is given to be 20 square units.
- The area of a rectangle is given by the formula:
[tex]\[ \text{Area} = \text{Width} \times \text{Height} \][/tex]
- Substituting the known values:
[tex]\[ 20 = 5 \times \text{Height} \][/tex]
- To find the height:
[tex]\[ \text{Height} = \frac{20}{5} = 4 \][/tex] units.
4. Find the Lower Vertices:
- Knowing the height of the rectangle, we can determine the y-coordinates of the lower vertices, as they will be 4 units down from the upper vertices.
- Lower-left corner:
- x-coordinate: Same as the upper-left, which is [tex]\(-1\)[/tex].
- y-coordinate: [tex]\(7 - 4 = 3\)[/tex].
- Lower-left corner is at [tex]\((-1, 3)\)[/tex].
- Lower-right corner:
- x-coordinate: Same as the upper-right, which is [tex]\(4\)[/tex].
- y-coordinate: [tex]\(7 - 4 = 3\)[/tex].
- Lower-right corner is at [tex]\((4, 3)\)[/tex].
5. Coordinates of All Vertices:
- Upper-left: [tex]\((-1, 7)\)[/tex]
- Upper-right: [tex]\((4, 7)\)[/tex]
- Lower-left: [tex]\((-1, 3)\)[/tex]
- Lower-right: [tex]\((4, 3)\)[/tex]
### Drawing the Rectangle:
Using the calculated coordinates, plot the rectangle on the coordinate plane:
1. Start by marking the upper-left corner at [tex]\((-1, 7)\)[/tex].
2. Mark the upper-right corner at [tex]\((4, 7)\)[/tex].
3. Mark the lower-left corner at [tex]\((-1, 3)\)[/tex].
4. Mark the lower-right corner at [tex]\((4, 3)\)[/tex].
Connect the dots to form a rectangle.
Here's a rough visualization:
```
y
8 |
7 | (4, 7)
6 |
5 |
4 |
3 | (4, 3)
2 |
1 |
0 |
- +
-1 0 1 2 3 4 x
```
This rectangle sits on the coordinate plane with the vertices as calculated.
### Step-by-Step Solution
1. Given Coordinates of the Upper Veritices:
- Upper-left corner is at [tex]\((-1, 7)\)[/tex].
- Upper-right corner is at [tex]\((4, 7)\)[/tex].
2. Calculate the Width:
- The width of the rectangle is the horizontal distance between the upper-left and upper-right vertices.
- The x-coordinate of the upper-left corner is [tex]\(-1\)[/tex].
- The x-coordinate of the upper-right corner is [tex]\(4\)[/tex].
- Width = [tex]\(4 - (-1) = 5\)[/tex] units.
3. Calculate the Height:
- The area of the rectangle is given to be 20 square units.
- The area of a rectangle is given by the formula:
[tex]\[ \text{Area} = \text{Width} \times \text{Height} \][/tex]
- Substituting the known values:
[tex]\[ 20 = 5 \times \text{Height} \][/tex]
- To find the height:
[tex]\[ \text{Height} = \frac{20}{5} = 4 \][/tex] units.
4. Find the Lower Vertices:
- Knowing the height of the rectangle, we can determine the y-coordinates of the lower vertices, as they will be 4 units down from the upper vertices.
- Lower-left corner:
- x-coordinate: Same as the upper-left, which is [tex]\(-1\)[/tex].
- y-coordinate: [tex]\(7 - 4 = 3\)[/tex].
- Lower-left corner is at [tex]\((-1, 3)\)[/tex].
- Lower-right corner:
- x-coordinate: Same as the upper-right, which is [tex]\(4\)[/tex].
- y-coordinate: [tex]\(7 - 4 = 3\)[/tex].
- Lower-right corner is at [tex]\((4, 3)\)[/tex].
5. Coordinates of All Vertices:
- Upper-left: [tex]\((-1, 7)\)[/tex]
- Upper-right: [tex]\((4, 7)\)[/tex]
- Lower-left: [tex]\((-1, 3)\)[/tex]
- Lower-right: [tex]\((4, 3)\)[/tex]
### Drawing the Rectangle:
Using the calculated coordinates, plot the rectangle on the coordinate plane:
1. Start by marking the upper-left corner at [tex]\((-1, 7)\)[/tex].
2. Mark the upper-right corner at [tex]\((4, 7)\)[/tex].
3. Mark the lower-left corner at [tex]\((-1, 3)\)[/tex].
4. Mark the lower-right corner at [tex]\((4, 3)\)[/tex].
Connect the dots to form a rectangle.
Here's a rough visualization:
```
y
8 |
7 | (4, 7)
6 |
5 |
4 |
3 | (4, 3)
2 |
1 |
0 |
- +
-1 0 1 2 3 4 x
```
This rectangle sits on the coordinate plane with the vertices as calculated.
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