Solved on Dec 06, 2023

Find the dimensions of a rectangle with upper-left coordinates $(-4,4)$ and upper-right coordinates $(4,4)$ that has a perimeter of 20 units.

STEP 1

Assumptions
1. The upper-left coordinates of the rectangle are $(-4,4)$ .
2. The upper-right coordinates of the rectangle are $(4,4)$ .
3. The perimeter of the rectangle is 20 units.
4. The rectangle sides are parallel to the coordinate axes.

STEP 2

Calculate the length of the rectangle using the coordinates of the upper-left and upper-right corners.
$Length = x_{right} - x_{left}$

STEP 3

Plug in the x-coordinates of the upper-right and upper-left corners to find the length.
$Length = 4 - (-4)$

STEP 4

Calculate the length of the rectangle.
$Length = 4 + 4 = 8 \text{ units}$

STEP 5

Use the perimeter formula for a rectangle to find the width. The perimeter $P$ of a rectangle is given by:
$P = 2 \times (Length + Width)$

STEP 6

Rearrange the formula to solve for the width.
$Width = \frac{P}{2} - Length$

STEP 7

Plug in the values for the perimeter and the length to calculate the width.
$Width = \frac{20}{2} - 8$

STEP 8

Calculate the width of the rectangle.
$Width = 10 - 8 = 2 \text{ units}$

STEP 9

Determine the coordinates of the lower-left and lower-right corners of the rectangle using the width.
The y-coordinate of the lower corners will be the y-coordinate of the upper corners minus the width.
$y_{lower} = y_{upper} - Width$

STEP 10

Calculate the y-coordinates of the lower-left and lower-right corners.
$y_{lower} = 4 - 2 = 2$

STEP 11

The x-coordinates of the lower corners are the same as the x-coordinates of the upper corners since the sides are parallel to the coordinate axes.
Thus, the lower-left corner is $(-4,2)$ and the lower-right corner is $(4,2)$ .

STEP 12

Now we can draw the rectangle on the coordinate plane using the coordinates of the four corners:
- Upper-left corner: $(-4,4)$ - Upper-right corner: $(4,4)$ - Lower-left corner: $(-4,2)$ - Lower-right corner: $(4,2)$
Connect these points with straight lines to form the rectangle.

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Find the dimensions of a rectangle with upper-left coordinates $(-4,4)$ and upper-right coordinates $(4,4)$ that has a perimeter of 20 units.

STEP 1

Assumptions
1. The upper-left coordinates of the rectangle are $(-4,4)$ .
2. The upper-right coordinates of the rectangle are $(4,4)$ .
3. The perimeter of the rectangle is 20 units.
4. The rectangle sides are parallel to the coordinate axes.

STEP 2

Calculate the length of the rectangle using the coordinates of the upper-left and upper-right corners.
$Length = x_{right} - x_{left}$

STEP 3

Plug in the x-coordinates of the upper-right and upper-left corners to find the length.
$Length = 4 - (-4)$

STEP 4

Calculate the length of the rectangle.
$Length = 4 + 4 = 8 \text{ units}$

STEP 5

Use the perimeter formula for a rectangle to find the width. The perimeter $P$ of a rectangle is given by:
$P = 2 \times (Length + Width)$

STEP 6

Rearrange the formula to solve for the width.
$Width = \frac{P}{2} - Length$

STEP 7

Plug in the values for the perimeter and the length to calculate the width.
$Width = \frac{20}{2} - 8$

STEP 8

Calculate the width of the rectangle.
$Width = 10 - 8 = 2 \text{ units}$

STEP 9

Determine the coordinates of the lower-left and lower-right corners of the rectangle using the width.
The y-coordinate of the lower corners will be the y-coordinate of the upper corners minus the width.
$y_{lower} = y_{upper} - Width$

STEP 10

Calculate the y-coordinates of the lower-left and lower-right corners.
$y_{lower} = 4 - 2 = 2$

STEP 11

The x-coordinates of the lower corners are the same as the x-coordinates of the upper corners since the sides are parallel to the coordinate axes.
Thus, the lower-left corner is $(-4,2)$ and the lower-right corner is $(4,2)$ .

STEP 12

Now we can draw the rectangle on the coordinate plane using the coordinates of the four corners:
- Upper-left corner: $(-4,4)$ - Upper-right corner: $(4,4)$ - Lower-left corner: $(-4,2)$ - Lower-right corner: $(4,2)$
Connect these points with straight lines to form the rectangle.

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Find the dimensions of a rectangle with upper-left coordinates (−4,4)(-4,4)(−4,4) and upper-right coordinates (4,4)(4,4)(4,4) that has a perimeter of 20 units.

STEP 1

Assumptions1. The upper-left coordinates of the rectangle are (−4,4)(-4,4)(−4,4).2. The upper-right coordinates of the rectangle are (4,4)(4,4)(4,4).3. The perimeter of the rectangle is 20 units.4. The rectangle sides are parallel to the coordinate axes.

STEP 2

Calculate the length of the rectangle using the coordinates of the upper-left and upper-right corners.Length=xright−xleftLength = x_{right} - x_{left}Length=xright​−xleft​

STEP 3

Plug in the x-coordinates of the upper-right and upper-left corners to find the length.Length=4−(−4)Length = 4 - (-4)Length=4−(−4)

STEP 4

Calculate the length of the rectangle.Length=4+4=8 unitsLength = 4 + 4 = 8 \text{ units}Length=4+4=8 units

STEP 5

Use the perimeter formula for a rectangle to find the width. The perimeter PPP of a rectangle is given by:P=2×(Length+Width)P = 2 \times (Length + Width)P=2×(Length+Width)

STEP 6

Rearrange the formula to solve for the width.Width=P2−LengthWidth = \frac{P}{2} - LengthWidth=2P​−Length

STEP 7

Plug in the values for the perimeter and the length to calculate the width.Width=202−8Width = \frac{20}{2} - 8Width=220​−8

STEP 8

Calculate the width of the rectangle.Width=10−8=2 unitsWidth = 10 - 8 = 2 \text{ units}Width=10−8=2 units

STEP 9

Determine the coordinates of the lower-left and lower-right corners of the rectangle using the width.The y-coordinate of the lower corners will be the y-coordinate of the upper corners minus the width.ylower=yupper−Widthy_{lower} = y_{upper} - Widthylower​=yupper​−Width

STEP 10

Calculate the y-coordinates of the lower-left and lower-right corners.ylower=4−2=2y_{lower} = 4 - 2 = 2ylower​=4−2=2

STEP 11

The x-coordinates of the lower corners are the same as the x-coordinates of the upper corners since the sides are parallel to the coordinate axes.Thus, the lower-left corner is (−4,2)(-4,2)(−4,2) and the lower-right corner is (4,2)(4,2)(4,2).

STEP 12

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Find the dimensions of a rectangle with upper-left coordinates $(-4,4)$ and upper-right coordinates $(4,4)$ that has a perimeter of 20 units.

Assumptions
1. The upper-left coordinates of the rectangle are $(-4,4)$ .
2. The upper-right coordinates of the rectangle are $(4,4)$ .
3. The perimeter of the rectangle is 20 units.
4. The rectangle sides are parallel to the coordinate axes.

Calculate the length of the rectangle using the coordinates of the upper-left and upper-right corners.
$Length = x_{right} - x_{left}$

Plug in the x-coordinates of the upper-right and upper-left corners to find the length.
$Length = 4 - (-4)$

Calculate the length of the rectangle.
$Length = 4 + 4 = 8 \text{ units}$

Use the perimeter formula for a rectangle to find the width. The perimeter $P$ of a rectangle is given by:
$P = 2 \times (Length + Width)$

Rearrange the formula to solve for the width.
$Width = \frac{P}{2} - Length$

Plug in the values for the perimeter and the length to calculate the width.
$Width = \frac{20}{2} - 8$

Calculate the width of the rectangle.
$Width = 10 - 8 = 2 \text{ units}$

Determine the coordinates of the lower-left and lower-right corners of the rectangle using the width.
The y-coordinate of the lower corners will be the y-coordinate of the upper corners minus the width.
$y_{lower} = y_{upper} - Width$

Calculate the y-coordinates of the lower-left and lower-right corners.
$y_{lower} = 4 - 2 = 2$

The x-coordinates of the lower corners are the same as the x-coordinates of the upper corners since the sides are parallel to the coordinate axes.
Thus, the lower-left corner is $(-4,2)$ and the lower-right corner is $(4,2)$ .