To find the distance between the points [tex]\(P = (-2, 5)\)[/tex] and [tex]\(Q = (1, 9)\)[/tex], we can use the distance formula for two points in a 2-dimensional coordinate plane. The distance formula is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the two points.
1. Assign the coordinates of point [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]:
Point [tex]\(P\)[/tex] has coordinates [tex]\((x_1, y_1) = (-2, 5)\)[/tex],
Point [tex]\(Q\)[/tex] has coordinates [tex]\((x_2, y_2) = (1, 9)\)[/tex].
2. Calculate the differences in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
[tex]\[ x_2 - x_1 = 1 - (-2) = 1 + 2 = 3 \][/tex]
[tex]\[ y_2 - y_1 = 9 - 5 = 4 \][/tex]
3. Substitute these differences into the distance formula:
[tex]\[ d = \sqrt{(3)^2 + (4)^2} \][/tex]
4. Simplify the expression inside the square root:
[tex]\[ d = \sqrt{9 + 16} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
5. Finally, take the square root of 25:
[tex]\[ d = 5 \][/tex]
Hence, the distance [tex]\(PQ\)[/tex] is [tex]\(5.0\)[/tex] units.
