To find the distance between the points [tex]\( P(2, 3) \)[/tex] and [tex]\( Q(5, -7) \)[/tex], we use the Euclidean distance formula. The formula to compute the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a Cartesian plane is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's apply this formula step-by-step for the points [tex]\( P(2, 3) \)[/tex] and [tex]\( Q(5, -7) \)[/tex]:
1. Identify the coordinates:
- [tex]\( x_1 = 2 \)[/tex]
- [tex]\( y_1 = 3 \)[/tex]
- [tex]\( x_2 = 5 \)[/tex]
- [tex]\( y_2 = -7 \)[/tex]
2. Calculate the difference between the [tex]\(x\)[/tex]-coordinates:
[tex]\[
x_2 - x_1 = 5 - 2 = 3
\][/tex]
3. Square this difference:
[tex]\[
(x_2 - x_1)^2 = 3^2 = 9
\][/tex]
4. Calculate the difference between the [tex]\(y\)[/tex]-coordinates:
[tex]\[
y_2 - y_1 = -7 - 3 = -10
\][/tex]
5. Square this difference:
[tex]\[
(y_2 - y_1)^2 = (-10)^2 = 100
\][/tex]
6. Sum the squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 9 + 100 = 109
\][/tex]
7. Find the square root of the sum to calculate the distance:
[tex]\[
d = \sqrt{109} \approx 10.4403
\][/tex]
Therefore, the distance between points [tex]\( P(2, 3) \)[/tex] and [tex]\( Q(5, -7) \)[/tex] is approximately [tex]\( 10.4403 \)[/tex].
