To plot the segment \( PQ \) with endpoints \( P(-1, -4) \) and \( Q(3, 1) \) on the coordinate plane, you would place point \( P \) at the coordinates \((-1, -4)\) in the fourth quadrant and point \( Q \) at the coordinates \((3, 1)\) in the first quadrant.
Next, we can find the length of segment \( PQ \) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (-1, -4) \) and \( (x_2, y_2) = (3, 1) \).
Plugging in the values:
\[ d = \sqrt{(3 - (-1))^2 + (1 - (-4))^2} \]
\[ d = \sqrt{(3 + 1)^2 + (1 + 4)^2} \]
\[ d = \sqrt{(4)^2 + (5)^2} \]
\[ d = \sqrt{16 + 25} \]
\[ d = \sqrt{41} \]
Thus, the length of segment \( PQ \) is \( \sqrt{41} \).
Next, we will find the midpoint \( M \) of segment \( PQ \) using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ M = \left( \frac{-1 + 3}{2}, \frac{-4 + 1}{2} \right) \]
\[ M = \left( \frac{2}{2}, \frac{-3}{2} \right) \]
\[ M = (1, -1.5) \]
Thus, the length of segment \( PQ \) is \( \sqrt{41} \), and the midpoint \( M \) is \( (1, -1.5) \).
So, to summarize:
- The length of \( PQ \) is \( \sqrt{41} \).
- The midpoint of \( PQ \) is \( M(1, -1.5) \).
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