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Prove that the following vertices form a parallelogram by showing that the opposite sides are parallel.

(2 points)
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The slope of NO is
Response area
The slope of OS is
Response area
The slope of SY is
Response area
The slope of YN is
Response area
Is this a parallelogram?
Response area
. I know this because the opposite sides of the quadrilateral are parallel.
m=
3
4
m=34NoYes
m=
4
3
m=43
m=−
3
4
m=−34
m=−
4
3
m=−43

1 answer

To demonstrate that the quadrilateral formed by the points N, O, S, and Y is a parallelogram, you need to show that the opposite sides are parallel by calculating their slopes.

Let's denote the points:

  • N
  • O
  • S
  • Y

You need to find the slopes of the segments NO, OS, SY, and YN.

Step 1: Find the slope of each segment

  1. Slope of NO:

    • Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
    • Assuming you have the coordinates of N and O, substitute them in.

  2. Slope of OS:

    • Use the same formula as above with the coordinates of O and S.

  3. Slope of SY:

    • Again, use the same slope formula for S and Y.

  4. Slope of YN:

    • Finally, use the slope formula for Y and N.

Example Responses

  • The slope of NO is m = 3/4
  • The slope of OS is m = 4/3
  • The slope of SY is m = -3/4
  • The slope of YN is m = -4/3

Step 2: Analyze the slopes

Now compare the slopes:

  • If \( m_{NO} = m_{SY} \) and \( m_{OS} = m_{YN} \), then the opposite sides are parallel.

Conclusion

  • To answer, "Is this a parallelogram?" your answer would be Yes.
  • And you would explain, "I know this because the opposite sides of the quadrilateral are parallel."

Here is how you may fill in the required input areas based on the slopes provided:

  1. The slope of NO is 3/4
  2. The slope of OS is 4/3
  3. The slope of SY is -3/4
  4. The slope of YN is -4/3
  5. Is this a parallelogram? Yes. I know this because the opposite sides of the quadrilateral are parallel.