In the right-angled triangles below, calculate the length of the sides that have not been given. Leave your answers in surd form where applicable.

For a right-angled triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
Given sides are 15 and 20 (hypotenuse). Let the unknown side be $y$.

The length of the side is $\boxed{5\sqrt{7}}$.

b)

Given sides are 24 and 25 (hypotenuse). Let the unknown side be $x$.

The length of the side is $\boxed{7}$.

1.2 A ladder with a length of 10m is placed at an angle against a wall. The bottom of the ladder is 2 meters away from the wall. How far up the wall will the ladder reach? Round off to two decimal places.

The ladder forms the hypotenuse ($c = 10$ m). The distance from the wall is one leg ($a = 2$ m). The height the ladder reaches up the wall is the other leg ($h$).
Apply the Pythagorean theorem: $a^2 + h^2 = c^2$.

The ladder will reach approximately $\boxed{9.80 \text{ m}}$ up the wall.

1.3 RQ is a vertical pole. The foot of the pole, Q, is on the same horizontal plane as P and S. The pole is anchored with cables RS and RP. The angle of depression from the top of the pole to the point S is $47^\circ$. RS is 20m and QS is 17m. $\angle RSQ = \theta$. PQ = QS.

a) Calculate the size of $\theta$.

The angle of depression from R to S is the angle between the horizontal line from R (let's call it RT) and the line of sight RS. So, $\angle TRS = 47^\circ$.

Since RQ is a vertical pole and QS is on a horizontal plane, RQ is perpendicular to QS. The horizontal line RT is parallel to QS.
Therefore, $\angle RSQ$ and $\angle TRS$ are alternate interior angles.

So, $\theta = \boxed{47^\circ}$.

b) Calculate the length of RQ.

$\triangle RQS$ is a right-angled triangle at Q (since RQ is vertical and QS is horizontal).
We are given the hypotenuse RS = 20 m and leg QS = 17 m. We need to find leg RQ.

The length of RQ is $\boxed{\sqrt{111} \text{ m}}$.

c) Calculate the length of RP.

We are given PQ = QS. Since QS = 17 m, then PQ = 17 m.
$\triangle RQP$ is a right-angled triangle at Q.
We have leg RQ = $\sqrt{111}$ m (from part b) and leg PQ = 17 m. We need to find the hypotenuse RP.

The length of RP is $\boxed{20 \text{ m}}$.