Given a right-angled triangle ABC, right-angled at B:

Step 1: Calculate the length of BC using the Pythagorean theorem.
Given a right-angled triangle ABC, right-angled at B:
$AB = 13 \text{ cm}$
$AC = 85 \text{ cm}$

The Pythagorean theorem states $AC^2 = AB^2 + BC^2$.

$$85^2 = 13^2 + BC^2$$ $$7225 = 169 + BC^2$$ $$BC^2 = 7225 - 169$$ $$BC^2 = 7056$$ $$BC = \sqrt{7056}$$ $$BC = 84 \text{ cm}$$

The length of BC is $\boxed{\text{84 cm}}$.

Step 2: Calculate the value of angle BAC.
Let $\theta = \angle BAC$.
We use the tangent ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB}$.

$$\tan(\theta) = \frac{84}{13}$$ $$\tan(\theta) \approx 6.461538$$

To find $\theta$, we take the inverse tangent:

$$\theta = \arctan(6.461538)$$ $$\theta \approx 81.2009^\circ$$

Correct to 2 decimal places, the value of angle BAC is $\boxed{81.20^\circ}$.

Step 1: Use the trigonometric identity $\cos A = \sin(90^\circ - A)$.
Given the equation $\cos(3x^\circ) = \sin(x-60^\circ)$.
We can rewrite $\cos(3x^\circ)$ as $\sin(90^\circ - 3x^\circ)$.
So, the equation becomes:

$$\sin(90^\circ - 3x^\circ) = \sin(x-60^\circ)$$

Step 2: Solve for $x$.
For $\sin A = \sin B$, the general solutions are $A = B + 360^\circ k$ or $A = 180^\circ - B + 360^\circ k$, where $k$ is an integer.
Since $x$ is an acute angle ($0^\circ < x < 90^\circ$), we consider the principal values.

Case 1: $90^\circ - 3x = x - 60^\circ$

$$90^\circ + 60^\circ = x + 3x$$ $$150^\circ = 4x$$ $$x = \frac{150^\circ}{4}$$ $$x = 37.5^\circ$$

This value of $x$ is acute. Let's verify:
$\cos(3 \times 37.5^\circ) = \cos(112.5^\circ)$
$\sin(37.5^\circ - 60^\circ) = \sin(-22.5^\circ)$
Since $\cos(112.5^\circ) = -\sin(22.5^\circ)$ and $\sin(-22.5^\circ) = -\sin(22.5^\circ)$, the equality holds.

Case 2: $90^\circ - 3x = 180^\circ - (x - 60^\circ)$

$$90^\circ - 3x = 180^\circ - x + 60^\circ$$ $$90^\circ - 3x = 240^\circ - x$$ $$90^\circ - 240^\circ = 3x - x$$ $$-150^\circ = 2x$$ $$x = -75^\circ$$

This value of $x$ is not acute, so it is rejected.

The value of $x$ is $\boxed{37.5^\circ}$.