a) Step 1: 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). Step 2: Constructing the mediator of PQ and finding S. • 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. Step 3: Constructing a circle. • 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. b) Step 1: Identify the components of the right-angled triangle formed by the broken tree. 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: $h_2$. Step 2: Use the Pythagorean theorem to find the length of the broken part ($h_2$). $$h_1^2 + d^2 = h_2^2$$ $$(12 \text{ ft})^2 + (13 \text{ ft})^2 = h_2^2$$ $$144 \text{ ft}^2 + 169 \text{ ft}^2 = h_2^2$$ $$313 \text{ ft}^2 = h_2^2$$ $$h_2 = \sqrt{313} \text{ ft}$$ Step 3: Calculate the total height of the tree before it broke. The total height $H$ is the sum of the standing part and the broken part. $$H = h_1 + h_2$$ $$H = 12 \text{ ft} + \sqrt{313} \text{ ft}$$ $$H \approx 12 \text{ ft} + 17.6918 \text{ ft}$$ $$H \approx 29.69 \text{ ft}$$ 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) Step 1: Write down the expression and given values. Expression: $\frac{8v^2 + 2vu}{7r} \times \frac{14r^2}{4v+u}$ Values: $v=2$, $u=-2$, $r=3$ Step 2: Simplify the expression by factoring. Factor out $2v$ from the numerator of the first fraction: $$8v^2 + 2vu = 2v(4v+u)$$ Substitute this back into the expression: $$\frac{2v(4v+u)}{7r} \times \frac{14r^2}{4v+u}$$ Cancel out the common term $(4v+u)$: $$\frac{2v}{7r} \times 14r^2$$ Simplify the constants and $r$ terms: $$\frac{2v \times 14r^2}{7r} = \frac{2v \times (2 \times 7) \times r \times r}{7r}$$ Cancel out $7r$: $$2v \times 2r = 4vr$$ Step 3: Substitute the given values into the simplified expression. $$4vr = 4(2)(3)$$ $$4(2)(3) = 8(3)$$ $$8(3) = 24$$ The evaluated expression is $\boxed{\text{}24}$. c.ii) Step 1: Write down the expression to factorize. $$12v^3 - 2u^2 + 8v^2u - 3vu$$ Step 2: Rearrange terms for grouping. Group terms with common factors: $$(12v^3 + 8v^2u) - (3vu + 2u^2)$$ Step 3: Factor out common terms from each group. From the first group $(12v^3 + 8v^2u)$, the common factor is $4v^2$: $$4v^2(3v + 2u)$$ From the second group $(-3vu - 2u^2)$, the common factor is $-u$: $$-u(3v + 2u)$$ Step 4: Combine the factored groups. $$(4v^2(3v + 2u)) - u(3v + 2u)$$ Now, $(3v + 2u)$ is a common factor for both terms: $$(3v + 2u)(4v^2 - u)$$ The factored expression is $\boxed{\text{}(3v + 2u)(4v^2 - u)}$.