Q. 1 - Precalculus Enhanced with Graphing Utilities

Question

Find the distance between the points P = (-1, 3) and Q = (4, -2). Find the midpoint of the line segment from P to Q .

Step-by-Step Solution

Answer

The distance between the points is $5\sqrt{2}$ and the midpoint is $\left(\frac{3}{2},\frac{1}{2}\right)$.

Step 1. Given information.

Find the distance between the points P = (-1, 3) and Q = (4, -2). Find the midpoint of the line segment from P to Q .

Step 2. Definition.

The distance between two points$T_1=\left(x_1, y_1\right), T_2=\left(x_2, y_2\right)$is given by:

$d(T_1,T_2)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

The mid point of line segment is given by: 

$\left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)$

Step 3. Apply the distance formula.

The distance between $P=(-1,3), Q=\left(4,-2\right):$

$$\begin{gathered} d(T_1,T_2)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2} \\ d(T_1,T_2)=\sqrt{{\left(4-(-1)\right)}^2+{\left(-2-3\right)}^2} \\ d(T_1,T_2)=\sqrt{{\left(5\right)}^2+{\left(-5\right)}^2} \\ d(T_1,T_2)=\sqrt{25+25} \\ d(T_1,T_2)=\sqrt{50} \\ d(T_1,T_2)=5\sqrt{2} \end{gathered}$$

Step 4. Apply the midpoint formula.

The midpoint R of the PQ:

$\left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)=\left(\frac{4-1}{2},\frac{-2+3}{2}\right)=\left(\frac{3}{2},\frac{1}{2}\right)$