Use the formulas for the area and perimeter of a rectangle to solve this problem.
Draw a rectangle that has:
an area of 42 square centimeters
a perimeter of 26 centimeters
The point on the grid represents one vertex of this rectangle.
Draw the sides of the rectangle along the grid lines. Each square in the grid has a side length of 1 centimeter.
To solve this problem, let's first denote the length of the rectangle as L and the width as W. We know that:
1. The area of the rectangle is given by the formula: Area = L × W.
2. The perimeter of the rectangle is given by the formula: Perimeter = 2 × (L + W).
We are given that the area is 42 square centimeters, and the perimeter is 26 centimeters:
1. 42 = L × W
2. 26 = 2 × (L + W)
Now we need to find the possible values that satisfy these two equations:
1. Factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.
2. Possible rectangle dimensions: (1, 42), (2, 21), (3, 14), (6, 7).
Now let's test these dimensions to see if any of them satisfy the perimeter equation:
1. Perimeter of (1, 42) = 2 × (1 + 42) = 86 ≠ 26
2. Perimeter of (2, 21) = 2 × (2 + 21) = 46 ≠ 26
3. Perimeter of (3, 14) = 2 × (3 + 14) = 34 ≠ 26
4. Perimeter of (6, 7) = 2 × (6 + 7) = 26 = 26
The dimensions (6, 7) satisfy both the required area and perimeter. So, the length is 6 centimeters and the width is 7 centimeters.
To draw the rectangle, place one vertex at the given point on the grid. Then, extend a horizontal line to the right by 6 centimeters and a vertical line downward by 7 centimeters. Complete the rectangle by connecting perpendicular lines.
Here's an example of what the drawing should look like:
```
6
+-+-+-+-+-+-+-+
|*| | | | | | |
+-+-+-+-+-+-+-+ 7
| | | | | | | |
+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+-+-+-+-+-+
| | | | | | | |
+-+-+-+-+-+-+-+
```
In the resulting image, the * symbol represents the starting vertex, and the +, - and | symbols represent the sides of the rectangle.