Question

what is the distance between a and b? a(-1,-4) b(4,9)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, point $A(-1,-4)$ and point $B(4,5)$, so $x_1=-1,y_1 = - 4,x_2=4,y_2 = 5$.

Step2: Calculate differences

First, find $x_2 - x_1$ and $y_2 - y_1$. $x_2 - x_1=4-(-1)=5$, $y_2 - y_1=5 - (-4)=9$.

Step3: Square the differences and sum

$(x_2 - x_1)^2=5^2 = 25$ and $(y_2 - y_1)^2=9^2 = 81$. Then $(x_2 - x_1)^2+(y_2 - y_1)^2=25 + 81=106$.

Step4: Take square - root

$d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{106}$.

Answer:

$\sqrt{106}$ (corresponding to the correct option in the multiple - choice question, though the options are not clearly readable in the provided image but the calculated value is $\sqrt{106}$)