Question

Plot the segment PQ with endpoints P(-1, -4) and Q (3, 1) on the coordinate plane. Then find the length and midpoint of PQ. Enter the midpoint coordinates as a decimal if necessary.

the length of PQ is ___
the midpoint PQ is M (___,___)

Answer 1

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 \( M(1, -1.5) \).