### Instruction: Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel?### Response: To find the distance Brenda has to travel, I can use the distance formula for the two segments she covers: from $(-4,5)$ to $(0,0)$ and from $(0,0)$ to $(5,-4)$.
The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints of the segment.
For the first segment, I have $(x_1,y_1)=(-4,5)$ and $(x_2,y_2)=(0,0)$.
Plugging these values into the distance formula, I get:

$d_1=\sqrt{(0-(-4))^2+(0-5)^2}$
Simplifying the expression, I get:

$d_1=\sqrt{16+25}=\sqrt{41}$
This is the distance Brenda travels from $(-4,5)$ to $(0,0)$.
For the second segment, I have $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(5,-4)$.
Plugging these values into the distance formula, I get:

$d_2=\sqrt{(5-0)^2+(-4-0)^2}$
Simplifying the expression, I get:

$d_2=\sqrt{25+16}=\sqrt{41}$
This is the distance Brenda travels from $(0,0)$ to $(5,-4)$.
To find the total distance Brenda travels, I need to add the two distances:

$d=d_1+d_2=\sqrt{41}+\sqrt{41}$