Question
Prove that the following vertices form a parallelogram by showing that the opposite sides are parallel.
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The slope of SA is Response area
The slope of AN is Response area
The slope of ND is Response area
The slope of DS is Response area
Is this a parallelogram? Response area. I know this because the opposite sides of the quadrilateral are parallel.
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The slope of SA is Response area
The slope of AN is Response area
The slope of ND is Response area
The slope of DS is Response area
Is this a parallelogram? Response area. I know this because the opposite sides of the quadrilateral are parallel.
Answers
GPT-4o mini
To prove that the given vertices form a parallelogram by showing that the opposite sides are parallel, we need to find the slopes of all four sides (SA, AN, ND, and DS) and determine if the opposite pairs have equal slopes.
1. **Calculate the slopes**:
- **Slope of SA**:
- If S = (x1, y1) and A = (x2, y2), then the slope (m) can be calculated as:
\[
m_{SA} = \frac{y2 - y1}{x2 - x1}
\]
- **Slope of AN**:
- If A = (x2, y2) and N = (x3, y3), then:
\[
m_{AN} = \frac{y3 - y2}{x3 - x2}
\]
- **Slope of ND**:
- If N = (x3, y3) and D = (x4, y4), then:
\[
m_{ND} = \frac{y4 - y3}{x4 - x3}
\]
- **Slope of DS**:
- If D = (x4, y4) and S = (x1, y1), then:
\[
m_{DS} = \frac{y1 - y4}{x1 - x4}
\]
2. **Compare slopes**:
- If \( m_{SA} = m_{ND} \) (opposite sides) and \( m_{AN} = m_{DS} \), then the opposite sides are parallel.
Now, fill in the following responses:
- The slope of SA is **m_{SA}** (insert the calculated value).
- The slope of AN is **m_{AN}** (insert the calculated value).
- The slope of ND is **m_{ND}** (insert the calculated value).
- The slope of DS is **m_{DS}** (insert the calculated value).
Finally, answer the last question:
- Is this a parallelogram? **Yes, I know this because the opposite sides of the quadrilateral are parallel.**
Please replace the placeholders with the actual slope values you calculated based on the coordinates of the points S, A, N, and D.
1. **Calculate the slopes**:
- **Slope of SA**:
- If S = (x1, y1) and A = (x2, y2), then the slope (m) can be calculated as:
\[
m_{SA} = \frac{y2 - y1}{x2 - x1}
\]
- **Slope of AN**:
- If A = (x2, y2) and N = (x3, y3), then:
\[
m_{AN} = \frac{y3 - y2}{x3 - x2}
\]
- **Slope of ND**:
- If N = (x3, y3) and D = (x4, y4), then:
\[
m_{ND} = \frac{y4 - y3}{x4 - x3}
\]
- **Slope of DS**:
- If D = (x4, y4) and S = (x1, y1), then:
\[
m_{DS} = \frac{y1 - y4}{x1 - x4}
\]
2. **Compare slopes**:
- If \( m_{SA} = m_{ND} \) (opposite sides) and \( m_{AN} = m_{DS} \), then the opposite sides are parallel.
Now, fill in the following responses:
- The slope of SA is **m_{SA}** (insert the calculated value).
- The slope of AN is **m_{AN}** (insert the calculated value).
- The slope of ND is **m_{ND}** (insert the calculated value).
- The slope of DS is **m_{DS}** (insert the calculated value).
Finally, answer the last question:
- Is this a parallelogram? **Yes, I know this because the opposite sides of the quadrilateral are parallel.**
Please replace the placeholders with the actual slope values you calculated based on the coordinates of the points S, A, N, and D.