$A$ ladder leans against a wall. It makes an angle of $45^{\circ}$ with the ground and reaches a height of $10 \, m$ on the wall. Find the length of the ladder and the distance between the lower end of the ladder and the base of the wall.

(N/A) Let the length of the ladder be $L$ and the distance from the wall be $x$.
Given: Height of the wall $h = 10 \, m$,Angle $\theta = 45^{\circ}$.
Using trigonometry:
$1.$ $\sin(45^{\circ}) = \frac{\text{Height}}{\text{Length}} = \frac{10}{L}$.
Since $\sin(45^{\circ}) = \frac{1}{\sqrt{2}}$,we have $\frac{1}{\sqrt{2}} = \frac{10}{L}$,which gives $L = 10\sqrt{2} \approx 14.14 \, m$.
$2.$ $\tan(45^{\circ}) = \frac{\text{Height}}{\text{Distance}} = \frac{10}{x}$.
Since $\tan(45^{\circ}) = 1$,we have $1 = \frac{10}{x}$,which gives $x = 10 \, m$.
Thus,the length of the ladder is $14.14 \, m$ and the distance from the wall is $10 \, m$.