Find the distance, [tex]d[/tex], of [tex]AB[/tex].

Given:
[tex]\[
\begin{array}{l}
A = (x_1, y_1) \\
B = (-6, -3)
\end{array}
\][/tex]

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

Calculate the distance [tex]d[/tex].

Round to the nearest unit.



Answer :

To find the distance [tex]\( d \)[/tex] between two points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], given their coordinates, we can use the distance formula. The distance formula in a 2-dimensional plane is:

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

In this problem, we are given:
- Point [tex]\( A \)[/tex] is at the origin, so [tex]\( A = (0, 0) \)[/tex].
- Point [tex]\( B \)[/tex] is at [tex]\( B = (-6, -3) \)[/tex].

To apply the distance formula, let's substitute [tex]\( x_1, y_1 \)[/tex] for the coordinates of point [tex]\( A \)[/tex] and [tex]\( x_2, y_2 \)[/tex] for the coordinates of point [tex]\( B \)[/tex]:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( y_1 = 0 \)[/tex]
- [tex]\( x_2 = -6 \)[/tex]
- [tex]\( y_2 = -3 \)[/tex]

Now, substitute these values into the formula:

[tex]\[ d = \sqrt{(-6 - 0)^2 + (-3 - 0)^2} \][/tex]

[tex]\[ d = \sqrt{(-6)^2 + (-3)^2} \][/tex]

Calculate the squares of the coordinates:

[tex]\[ (-6)^2 = 36 \][/tex]
[tex]\[ (-3)^2 = 9 \][/tex]

Then, add these results:

[tex]\[ d = \sqrt{36 + 9} \][/tex]

[tex]\[ d = \sqrt{45} \][/tex]

Next, we need to take the square root of 45:

[tex]\[ d \approx 6.7082 \][/tex]

Now, we need to round this value to the nearest cent (two decimal places):

[tex]\[ d \approx 6.71 \][/tex]

Therefore, the distance [tex]\( d \)[/tex] between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately [tex]\( 6.71 \)[/tex].