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Point [tex]$C$[/tex] has the coordinates [tex]$(-1, 4)$[/tex] and point [tex]$D$[/tex] has the coordinates [tex]$(2, 0)$[/tex]. What is the distance between points [tex]$C$[/tex] and [tex]$D$[/tex]?

[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

[tex]$\square$[/tex] units


Sagot :

To determine the distance between point [tex]\( C \)[/tex] with coordinates [tex]\((-1, 4)\)[/tex] and point [tex]\( D \)[/tex] with coordinates [tex]\( (2, 0) \)[/tex], we will use the distance formula. The distance formula is:

[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Here:
- [tex]\((x_1, y_1) = (-1, 4)\)[/tex] are the coordinates of point [tex]\( C \)[/tex],
- [tex]\((x_2, y_2) = (2, 0)\)[/tex] are the coordinates of point [tex]\( D \)[/tex].

Step-by-step procedure:

1. Calculate the difference in the x-coordinates ([tex]\(x_2 - x_1\)[/tex]):
[tex]\[ x_2 - x_1 = 2 - (-1) = 2 + 1 = 3 \][/tex]

2. Calculate the difference in the y-coordinates ([tex]\(y_2 - y_1\)[/tex]):
[tex]\[ y_2 - y_1 = 0 - 4 = -4 \][/tex]

3. Square the differences:
[tex]\[ (x_2 - x_1)^2 = 3^2 = 9 \][/tex]
[tex]\[ (y_2 - y_1)^2 = (-4)^2 = 16 \][/tex]

4. Sum the squares of the differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 9 + 16 = 25 \][/tex]

5. Take the square root of the sum to find the distance:
[tex]\[ d = \sqrt{25} = 5 \][/tex]

Thus, the distance between points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
units.