Question Number 77272 by jagoll last updated on 05/Jan/20
$$\mathrm{given}\:\mathrm{T}.\mathrm{ABCD}\:\mathrm{is}\:\mathrm{pyramid}\: \\ $$$$\mathrm{with}\:\mathrm{AB}\:=\:\mathrm{BC}\:=\:\mathrm{8}\:\mathrm{and}\:\mathrm{AT}\:=\:\mathrm{6} \\ $$$$\:.\:\mathrm{P}\:\mathrm{is}\:\mathrm{midpoint}\:\mathrm{BC},\:\mathrm{Q}\:\mathrm{is}\:\mathrm{midpoint}\:\mathrm{AT}. \\ $$$$\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{between}\:\mathrm{TP}\:\mathrm{and} \\ $$$$\mathrm{PQ}\:\mathrm{then}\:\mathrm{cos}\:\alpha\:\mathrm{is}\:… \\ $$
Commented by john santu last updated on 05/Jan/20
Commented by john santu last updated on 05/Jan/20
$${look}\:{this}\:{picture}\:{sir} \\ $$
Commented by jagoll last updated on 05/Jan/20
$$\mathrm{thanks}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by mr W last updated on 05/Jan/20
Commented by mr W last updated on 05/Jan/20
$${AP}=\sqrt{\mathrm{4}^{\mathrm{2}} +\mathrm{8}^{\mathrm{2}} }=\mathrm{4}\sqrt{\mathrm{5}} \\ $$$${TP}=\sqrt{\mathrm{6}^{\mathrm{2}} −\mathrm{4}^{\mathrm{2}} }=\mathrm{2}\sqrt{\mathrm{5}} \\ $$$${AQ}={QT}=\frac{\mathrm{6}}{\mathrm{2}}=\mathrm{3} \\ $$$$\mathrm{cos}\:{A}=\frac{\mathrm{6}^{\mathrm{2}} +\left(\mathrm{4}\sqrt{\mathrm{5}}\right)^{\mathrm{2}} −\left(\mathrm{2}\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }{\mathrm{2}×\mathrm{6}×\mathrm{4}\sqrt{\mathrm{5}}}=\frac{\mathrm{2}}{\:\sqrt{\mathrm{5}}} \\ $$$${PQ}^{\mathrm{2}} =\mathrm{3}^{\mathrm{2}} +\left(\mathrm{4}\sqrt{\mathrm{5}}\right)^{\mathrm{2}} −\mathrm{2}×\mathrm{3}×\mathrm{4}\sqrt{\mathrm{5}}×\mathrm{cos}\:{A}=\mathrm{41} \\ $$$$\mathrm{cos}\:\alpha=\frac{\mathrm{41}+\left(\mathrm{2}\sqrt{\mathrm{5}}\right)^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }{\mathrm{2}×\sqrt{\mathrm{41}}×\mathrm{2}\sqrt{\mathrm{5}}}=\frac{\mathrm{13}}{\:\sqrt{\mathrm{205}}}=\frac{\mathrm{13}\sqrt{\mathrm{205}}}{\mathrm{205}} \\ $$
Commented by john santu last updated on 05/Jan/20
$${good}\:{sir}.\:{i}'{m}\:{to}\:{solve}\:{by}\:{vector}\:.\:{what}\:{result}\:{you}\:{got}\:{sir}? \\ $$
Commented by john santu last updated on 05/Jan/20
$${i}\:{got}\:\mathrm{cos}\:\alpha\:=\:\frac{\mathrm{13}}{\:\sqrt{\mathrm{205}\:}}\:{sir} \\ $$
Commented by john santu last updated on 05/Jan/20