To find the distance between the points [tex]\( A(4, -3) \)[/tex] and [tex]\( B(3, 5) \)[/tex] on the coordinate plane, we'll use the Distance Formula. The Distance Formula is defined as:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] are the coordinates of point [tex]\( A \)[/tex], and [tex]\((x_2, y_2)\)[/tex] are the coordinates of point [tex]\( B \)[/tex].
Given:
[tex]\( x_1 = 4 \)[/tex]
[tex]\( y_1 = -3 \)[/tex]
[tex]\( x_2 = 3 \)[/tex]
[tex]\( y_2 = 5 \)[/tex]
Substitute these values into the Distance Formula:
[tex]\[ \text{Distance} = \sqrt{(3 - 4)^2 + (5 - (-3))^2} \][/tex]
Simplify inside the parentheses first:
[tex]\[ \text{Distance} = \sqrt{(3 - 4)^2 + (5 + 3)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(-1)^2 + (8)^2} \][/tex]
Next, compute the squares:
[tex]\[ \text{Distance} = \sqrt{1 + 64} \][/tex]
Then, add the values inside the square root:
[tex]\[ \text{Distance} = \sqrt{65} \][/tex]
Finally, take the square root of 65:
[tex]\[ \text{Distance} \approx 8.06225774829855 \][/tex]
Now, rounding this to the nearest tenth:
[tex]\[ \text{Distance} \approx 8.1 \][/tex]
So, the distance between points [tex]\( A(4, -3) \)[/tex] and [tex]\( B(3, 5) \)[/tex] is approximately [tex]\( 8.1 \)[/tex] units when rounded to the nearest tenth.
