Problem
Exercice 4 (5 poiats) On considère la pjramide \( S A B C D \) dont la base est le rectangle \( A B C D \) et de hauteur [SHI, On doane: \( \mathrm{AB}=8 \mathrm{~cm}, \mathrm{AD}=4 \mathrm{~cm}, \widehat{\mathrm{SHC}}=90^{\circ} \) et \( \widehat{\mathrm{SAC}}=60^{\circ} \). 1) Montrer qua \( A C=4 \sqrt{5} \mathrm{cmm} \), [juis calculer \( A \mathrm{~A}_{\text {. }} \). 2) Léduire que \( \mathrm{SH}=2 \sqrt{15} \mathrm{~cm} \) 3) Caiculer le volume de la pyramide SABCD. 4) La pyramide \( \mathrm{SABCD} \) est-elle régulière? Justifier.
Solution
Step 1 ：1) Use Pythagorean theorem on triangle SAC: \(AC^2 = SA^2 + SC^2\), given \(SA = 4, \widehat{SAC}=60^{\circ}\), find AC.
Step 2 ：\(AC = 4\sqrt{5}\) cm and \(A \mathrm{A}_{.} = \frac{1}{2} \times AB \times AD = 16 \mathrm{~cm^{2}}\)
Step 3 ：2) Use Pythagorean theorem on triangle SHC: \(SH^2 = SC^2 - HC^2\), find SH.
Step 4 ：\(SH = 2\sqrt{15}\) cm
Step 5 ：3) Calculate volume of pyramid SABCD: \(V = \frac{1}{3} \times A \mathrm{A}_{.} \times SH\)
Step 6 ：V = 32\sqrt{15} \mathrm{~cm^{3}}
Step 7 ：4) The pyramid is not regular because the base is a rectangle, not a square.
