4. Tangents are drawn from point $T$ to a circle with a radius of $4 \mathrm{~cm}$. Calculate the length of the arc of the circle visible from point $T$, if the tangents enclose an angle of $45^{\circ}$.
First method:

Sketch:

$1 \text{ POINT}$

![](https://cdn.mathpix.com/cropped/2024_05_30_4e3bea37fa3879b468d2g-3.jpg?height=728&width=945&top_left_y=1109&top_left_x=561)

The tangents are perpendicular to the radii $\overline{A S}$ and $\overline{B S}$

$1 \text{ POINT}$

Angles $\angle A S B$ and $\angle A T B$ are angles with perpendicular legs, one of which is acute and the other is obtuse, so:

$\alpha=180^{\circ}-45^{\circ}=135^{\circ}$

$1 \text{ POINT}$

The length of the arc is calculated using the formula $l=\frac{r \cdot \pi \cdot \alpha}{180^{\circ}}$, so

$l=\frac{4 \cdot \pi \cdot 135^{\circ}}{180^{\circ}}=3 \pi$

$2 \text{ POINTS}$

The length of the arc seen from point $T$ is $l=3 \pi \text{ cm}$.

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