Answer
$\sqrt{29}$ miles
Work Step by Step
Let's consider Li starts from point $A(0,0)$. She drives $2$ miles north and reaches the point $B(0,2)$. Then she moves $5$ miles east, which means point $C(5,2)$. Then she goes $4$ miles south reaching point $D(5,-2)$.
We have to calculate the distance from $A$ to $D$ using the distance formula: $$\begin{align*} d_{AD}&=\sqrt{(x_D-x_A)^2+(y_D-x_A)^2}\\ &=\sqrt{(5-0)^2+(-2-0)^2}\\ &=\sqrt{29}\approx 5.385. \end{align*}$$
The distance between Li's home and her office is $\sqrt{29}$ miles.
