Question
A rectangle is shown in the figure below. What is the perimeter? Units are in inches.
graph shows 4 points (3,4) (7,2) (-3,1) (1,1)
(1 point)
Responses
155–√ in
155–√ in
105–√ in
105–√ in
265–√ in
265–√ in
55–√ in
55–√ in
graph shows 4 points (3,4) (7,2) (-3,1) (1,1)
(1 point)
Responses
155–√ in
155–√ in
105–√ in
105–√ in
265–√ in
265–√ in
55–√ in
55–√ in
Answers
GPT-4o mini
To calculate the perimeter of the rectangle given the points \((3,4)\), \((7,2)\), \((-3,1)\), and \((1,1)\), we first need to verify that these points indeed form a rectangle and find the dimensions.
### Step 1: Identify the lengths of the sides
1. Calculate the distances between all pairs of points to identify pairs that represent the opposite corners of the rectangle.
2. Use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Distance Calculations:
- From \((3,4)\) to \((7,2)\):
\[
d = \sqrt{(7 - 3)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- From \((3,4)\) to \((-3,1)\):
\[
d = \sqrt{(-3 - 3)^2 + (1 - 4)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]
- From \((3,4)\) to \((1,1)\):
\[
d = \sqrt{(1 - 3)^2 + (1 - 4)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- From \((7,2)\) to \((-3,1)\):
\[
d = \sqrt{(-3 - 7)^2 + (1 - 2)^2} = \sqrt{100 + 1} = \sqrt{101}
\]
- From \((7,2)\) to \((1,1)\):
\[
d = \sqrt{(1 - 7)^2 + (1 - 2)^2} = \sqrt{36 + 1} = \sqrt{37}
\]
- From \((-3,1)\) to \((1,1)\):
\[
d = \sqrt{(1 + 3)^2 + (1 - 1)^2} = \sqrt{16} = 4
\]
### Step 2: Identify Opposite Corners
- The points pair \( \{(3, 4), (1, 1)\} \) and \( \{(7, 2), (-3, 1)\} \) have the calculated distances which will indicate opposite sides of the rectangle.
### Step 3: Calculate Length and Width
- The distance between \((3,4)\) to \((1,1)\) can represent one dimension (height), and the distance between \((7,2)\) to \((-3,1)\) can represent the other (width).
### Step 4: Calculate the perimeter
- Perimeter \(P\) of a rectangle is given by:
\[
P = 2(\text{length} + \text{width})
\]
Assuming:
- Length from \((3,4)\) to \((1,1)\) is height = \(\sqrt{13}\)
- Width from \((7,2)\) to \((-3,1)\) is width = \(4\)
Total dimensions:
\[
P = 2( \sqrt{13} + 4) \approx 2(\sqrt{13} + 4)
\]
### Final Calculation
Calculating the actual values leads us to varied choices for the perimeter which should check against the presented options. Given that none of the calculations led to a whole number, and all seem to round around numerical choices, therefore none fits.
To finalize based on calculation:
- **The nearest match from the options is ** \( 155 - \sqrt{} in\)
Hence, please ensure there is no discrepancy or if further points were suggested to derive.
### Step 1: Identify the lengths of the sides
1. Calculate the distances between all pairs of points to identify pairs that represent the opposite corners of the rectangle.
2. Use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Distance Calculations:
- From \((3,4)\) to \((7,2)\):
\[
d = \sqrt{(7 - 3)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- From \((3,4)\) to \((-3,1)\):
\[
d = \sqrt{(-3 - 3)^2 + (1 - 4)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]
- From \((3,4)\) to \((1,1)\):
\[
d = \sqrt{(1 - 3)^2 + (1 - 4)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- From \((7,2)\) to \((-3,1)\):
\[
d = \sqrt{(-3 - 7)^2 + (1 - 2)^2} = \sqrt{100 + 1} = \sqrt{101}
\]
- From \((7,2)\) to \((1,1)\):
\[
d = \sqrt{(1 - 7)^2 + (1 - 2)^2} = \sqrt{36 + 1} = \sqrt{37}
\]
- From \((-3,1)\) to \((1,1)\):
\[
d = \sqrt{(1 + 3)^2 + (1 - 1)^2} = \sqrt{16} = 4
\]
### Step 2: Identify Opposite Corners
- The points pair \( \{(3, 4), (1, 1)\} \) and \( \{(7, 2), (-3, 1)\} \) have the calculated distances which will indicate opposite sides of the rectangle.
### Step 3: Calculate Length and Width
- The distance between \((3,4)\) to \((1,1)\) can represent one dimension (height), and the distance between \((7,2)\) to \((-3,1)\) can represent the other (width).
### Step 4: Calculate the perimeter
- Perimeter \(P\) of a rectangle is given by:
\[
P = 2(\text{length} + \text{width})
\]
Assuming:
- Length from \((3,4)\) to \((1,1)\) is height = \(\sqrt{13}\)
- Width from \((7,2)\) to \((-3,1)\) is width = \(4\)
Total dimensions:
\[
P = 2( \sqrt{13} + 4) \approx 2(\sqrt{13} + 4)
\]
### Final Calculation
Calculating the actual values leads us to varied choices for the perimeter which should check against the presented options. Given that none of the calculations led to a whole number, and all seem to round around numerical choices, therefore none fits.
To finalize based on calculation:
- **The nearest match from the options is ** \( 155 - \sqrt{} in\)
Hence, please ensure there is no discrepancy or if further points were suggested to derive.