Question Number 111278 by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{Towns}\:\mathrm{A},\mathrm{B},\mathrm{C}\:\mathrm{and}\:\mathrm{D}\:\mathrm{are}\:\mathrm{located}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{whose}\:\mathrm{area}\:\mathrm{is} \\ $$$$\mathrm{1000km}^{\mathrm{2}} .\:\mathrm{There}\:\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{highway}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{square}\:\mathrm{but}\:\mathrm{not}\:\mathrm{through}\:\mathrm{any}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{towns}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squares} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{distances}\:\mathrm{of}\:\mathrm{the}\:\mathrm{towns}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{highway}. \\ $$
Answered by ajfour last updated on 03/Sep/20
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{Solution}\:\mathrm{please}? \\ $$
Commented by ajfour last updated on 03/Sep/20
$${p}+{q}={a}\mathrm{sin}\:\theta \\ $$$${p}−{q}={a}\mathrm{cos}\:\theta \\ $$$${d}_{{A}} ^{\mathrm{2}} +{d}_{{B}} ^{\mathrm{2}} +{d}_{{C}} ^{\mathrm{2}} +{d}_{{D}} ^{\mathrm{2}} =\:{q}^{\mathrm{2}} +{p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{p}^{\mathrm{2}} \\ $$$$\:\:\:\:=\:\mathrm{2}\left({p}^{\mathrm{2}} +{q}^{\mathrm{2}} \right)\:=\:\left({a}\mathrm{sin}\:\theta\right)^{\mathrm{2}} +\left({a}\mathrm{cos}\:\theta\right)^{\mathrm{2}} \\ $$$$\:\:\:\:=\:{a}^{\mathrm{2}} \:=\:\mathrm{1000}{km}^{\mathrm{2}} \:. \\ $$
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
$$\mathrm{Ok}.\:\mathrm{Thanks}.\:\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{that}. \\ $$