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].
