To find the distance between the points [tex]\( R(-11, 7) \)[/tex] and [tex]\( T(9, -15) \)[/tex], we need to use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the coordinate plane is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's identify our coordinates:
- For point [tex]\( R \)[/tex], [tex]\( x_1 = -11 \)[/tex] and [tex]\( y_1 = 7 \)[/tex]
- For point [tex]\( T \)[/tex], [tex]\( x_2 = 9 \)[/tex] and [tex]\( y_2 = -15 \)[/tex]
First, we calculate the differences in the x-coordinates and y-coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 9 - (-11) = 9 + 11 = 20
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = -15 - 7 = -15 - 7 = -22
\][/tex]
Next, we plug these differences into the distance formula:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(20)^2 + (-22)^2}
\][/tex]
Let's compute the squares of these differences:
[tex]\[
(20)^2 = 400
\][/tex]
[tex]\[
(-22)^2 = 484
\][/tex]
Now, we add these squared differences:
[tex]\[
400 + 484 = 884
\][/tex]
Finally, we take the square root of this sum to find the distance:
[tex]\[
d = \sqrt{884} \approx 29.732137494637012
\][/tex]
Therefore, the distance between the points [tex]\( R(-11, 7) \)[/tex] and [tex]\( T(9, -15) \)[/tex] is approximately [tex]\( 29.732 \)[/tex].
