To find the distance between two points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] given their coordinates [tex]\( (-1, 4) \)[/tex] and [tex]\( (2, 0) \)[/tex] respectively, we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here's a step-by-step solution:
1. Identify the coordinates of points [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- Point [tex]\( C \)[/tex] has coordinates [tex]\( (x_1, y_1) = (-1, 4) \)[/tex].
- Point [tex]\( D \)[/tex] has coordinates [tex]\( (x_2, y_2) = (2, 0) \)[/tex].
2. Calculate the difference in x-coordinates ([tex]\( \Delta x \)[/tex]):
[tex]\[
\Delta x = x_2 - x_1 = 2 - (-1) = 2 + 1 = 3
\][/tex]
3. Calculate the difference in y-coordinates ([tex]\( \Delta y \)[/tex]):
[tex]\[
\Delta y = y_2 - y_1 = 0 - 4 = -4
\][/tex]
4. Square the differences:
[tex]\[
(\Delta x)^2 = 3^2 = 9
\][/tex]
[tex]\[
(\Delta y)^2 = (-4)^2 = 16
\][/tex]
5. Sum the squares of the differences:
[tex]\[
(\Delta x)^2 + (\Delta y)^2 = 9 + 16 = 25
\][/tex]
6. Calculate the distance by taking the square root of the sum:
[tex]\[
d = \sqrt{25} = 5
\][/tex]
So, the distance between points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] is [tex]\( 5 \)[/tex] units.
