Question
Suppose a parallelogram is drawn on a coordinate plane. Using verticies with coordinates (2,3), (6,3), (7,7), and (3,7), compute the perimeter. Round your answer to the nearest tenth.
The perimeter is approximately __ units
a. 20.0
b. 16.0
c. 22.6
d. 18.9
The perimeter is approximately __ units
a. 20.0
b. 16.0
c. 22.6
d. 18.9
Answers
GPT-4o mini
To compute the perimeter of the given parallelogram with vertices at coordinates (2,3), (6,3), (7,7), and (3,7), we can find the lengths of two pairs of opposite sides.
1. **Finding the length of the side between (2,3) and (6,3):**
\[
\text{Length} = \sqrt{(6 - 2)^2 + (3 - 3)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4
\]
2. **Finding the length of the side between (3,7) and (7,7):**
\[
\text{Length} = \sqrt{(7 - 3)^2 + (7 - 7)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4
\]
Both pairs of opposite sides in a parallelogram are equal, so both horizontal sides (2,3) to (6,3) and (3,7) to (7,7) are 4 units long.
3. **Finding the length of the vertical side between (2,3) and (3,7):**
\[
\text{Length} = \sqrt{(3 - 2)^2 + (7 - 3)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1231
\]
4. **Finding the length on the opposite vertical side between (6,3) and (7,7):**
\[
\text{Length} = \sqrt{(7 - 6)^2 + (7 - 3)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1231
\]
Now we can calculate the perimeter of the parallelogram:
\[
\text{Perimeter} = 2 \times (\text{Length of horizontal side} + \text{Length of vertical side}) = 2 \times (4 + \sqrt{17}) \approx 2 \times (4 + 4.1231) \approx 2 \times 8.1231 \approx 16.2462
\]
Rounding to the nearest tenth gives us 16.2.
However, when comparing with provided options, let's state the definitive values:
1. Horizontal Sides: \(4\) units each
2. Vertical Sides: \(\sqrt{17} \approx 4.1231\) units each
Calculating Perimeter:
\[
\text{Perimeter} = 2 \times (4 + 4.1231) \approx 16.2
\]
The closest answer aligns with option b:
b. \(16.0\)
Hence, the perimeter of the parallelogram is approximately **16.0 units**.
1. **Finding the length of the side between (2,3) and (6,3):**
\[
\text{Length} = \sqrt{(6 - 2)^2 + (3 - 3)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4
\]
2. **Finding the length of the side between (3,7) and (7,7):**
\[
\text{Length} = \sqrt{(7 - 3)^2 + (7 - 7)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4
\]
Both pairs of opposite sides in a parallelogram are equal, so both horizontal sides (2,3) to (6,3) and (3,7) to (7,7) are 4 units long.
3. **Finding the length of the vertical side between (2,3) and (3,7):**
\[
\text{Length} = \sqrt{(3 - 2)^2 + (7 - 3)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1231
\]
4. **Finding the length on the opposite vertical side between (6,3) and (7,7):**
\[
\text{Length} = \sqrt{(7 - 6)^2 + (7 - 3)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1231
\]
Now we can calculate the perimeter of the parallelogram:
\[
\text{Perimeter} = 2 \times (\text{Length of horizontal side} + \text{Length of vertical side}) = 2 \times (4 + \sqrt{17}) \approx 2 \times (4 + 4.1231) \approx 2 \times 8.1231 \approx 16.2462
\]
Rounding to the nearest tenth gives us 16.2.
However, when comparing with provided options, let's state the definitive values:
1. Horizontal Sides: \(4\) units each
2. Vertical Sides: \(\sqrt{17} \approx 4.1231\) units each
Calculating Perimeter:
\[
\text{Perimeter} = 2 \times (4 + 4.1231) \approx 16.2
\]
The closest answer aligns with option b:
b. \(16.0\)
Hence, the perimeter of the parallelogram is approximately **16.0 units**.