Constructing triangle PQR.

• Draw a line segment PQ of length 8 cm.
• At point P, construct an angle of $45^\circ$ with PQ. Draw a ray from P.
• Since $\angle QRP = 90^\circ$ and $\angle QPR = 45^\circ$, the third angle $\angle PQR = 180^\circ - 90^\circ - 45^\circ = 45^\circ$.
• At point Q, construct an angle of $45^\circ$ with PQ. Draw a ray from Q.
• The intersection of the two rays from P and Q is point R.
• Measure the length of QR. (This measurement cannot be provided without physical construction).

• With P as the center and a radius greater than half the length of PQ (i.e., greater than 4 cm), draw arcs above and below PQ.
• With Q as the center and the same radius, draw arcs that intersect the previously drawn arcs.
• Draw a straight line connecting the two intersection points of the arcs. This line is the perpendicular bisector (mediator) of PQ.
• Mark the point where this perpendicular bisector intersects the line segment PR as S.

• Place the compass point at S (the center).
• Adjust the compass opening to a radius of 3 cm.
• Draw a circle with center S and radius 3 cm.

The height where the tree broke is one leg of the right triangle: $h_1 = 12$ ft.
The distance from the base of the tree to where the top touches the ground is the other leg: $d = 13$ ft.
The broken part of the tree forms the hypotenuse: $h_2$.

The total height $H$ is the sum of the standing part and the broken part.

The height of the tree before the wind broke it was $\boxed{\text{}12 + \sqrt{313} \text{ ft} \approx 29.69 \text{ ft}}$.

c.i)

Expression: $\frac{8v^2 + 2vu}{7r} \times \frac{14r^2}{4v+u}$
Values: $v=2$, $u=-2$, $r=3$

Factor out $2v$ from the numerator of the first fraction:

Substitute this back into the expression:

Cancel out the common term $(4v+u)$:

Simplify the constants and $r$ terms:

Cancel out $7r$:

The evaluated expression is $\boxed{\text{}24}$.

c.ii)

Group terms with common factors:

From the first group $(12v^3 + 8v^2u)$, the common factor is $4v^2$:

From the second group $(-3vu - 2u^2)$, the common factor is $-u$:

Now, $(3v + 2u)$ is a common factor for both terms:

The factored expression is $\boxed{\text{}(3v + 2u)(4v^2 - u)}$.