4. ABCD is a rectangle in which diagonal AC bisects \( \angle \mathrm{A} \) as well as \( \angle \mathrm{C} \). Show that: (i) ABCD is a square (ii) diagonal BD bisects \( \angle \mathrm{B} \) as well as \( \angle \mathrm{D} \).

Volume of a Tank A civil engineering firm is tasked with designing a new water storage tank for a municipal water supply system. The engineers want the tank to have an efficient shape that maximizes storage capacity while minimizing the amount of construction material used. The tank's cross-section will be shaped by revolving the regions bounded by the lines and curves, \( x=\sqrt{2 \sin ^{2} y} \) \( 0 \leq y \leq \frac{\pi}{2}, x=0 \) about the \( y \)-axis. The goal is to determine the volume of the tank that will be created by this revolution around the \( y \)-axis. 4.1 Source: Mafemba, J. (2024) a. Calculate the volume of the solid generated by revolving the regions bounded by the lines and curves mentioned in the scenario above

2. \( \tan (\theta)+\cot (\theta) \equiv \sec (\theta) \cdot \csc (\theta) \)

B is the midpoint of segment AD. C is the midpoint of segment BD. located at \( (-1,-6) \) and \( D \) is located at \( (3,0) \). What are the coordinate: Draw and label a picture to help. ( \( -1,3) \) \( O(2,-1.5) \) \( O(-3,1) \) \( O(-4,3) \)

Solve each system of equations by elimination. \[ \begin{aligned} x-5 y & =26 \\ -7 x+5 y & =-2\end{aligned} \]

In un rettangolo il perimetro è di 40 cm e l'al terra èil triplo della base. Calcola la lunghezz detle due dimensioni. \( \quad \) [ 5 cm ; 15 cm