To find the distance between two points \( A(5,8) \) and \( B(-3,4) \) in a Cartesian plane, you can use the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, \((x_1, y_1)\) are the coordinates of point \( A \) and \((x_2, y_2)\) are the coordinates of point \( B \).
1. Identify the coordinates of the points:
- Point \( A \) has coordinates \((5, 8)\).
- Point \( B \) has coordinates \((-3, 4)\).
2. Substitute these coordinates into the distance formula:
[tex]\[
d = \sqrt{(-3 - 5)^2 + (4 - 8)^2}
\][/tex]
3. Calculate the differences inside the parentheses:
[tex]\[
x_2 - x_1 = -3 - 5 = -8
\][/tex]
[tex]\[
y_2 - y_1 = 4 - 8 = -4
\][/tex]
4. Substitute these differences back into the formula:
[tex]\[
d = \sqrt{(-8)^2 + (-4)^2}
\][/tex]
5. Square these differences:
[tex]\[
(-8)^2 = 64
\][/tex]
[tex]\[
(-4)^2 = 16
\][/tex]
6. Add the squared differences:
[tex]\[
64 + 16 = 80
\][/tex]
7. Finally, take the square root of the sum to find the distance:
[tex]\[
d = \sqrt{80}
\][/tex]
8. Simplifying the square root of 80:
[tex]\[
d = \sqrt{16 \times 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}
\][/tex]
9. Approximating further to a numerical value, we get:
[tex]\[
d \approx 8.94427190999916
\][/tex]
Therefore, the distance between the points [tex]\( A(5,8) \)[/tex] and [tex]\( B(-3,4) \)[/tex] is approximately [tex]\( 8.94427190999916 \)[/tex] units.
