Question

ABCD is a tetrahedron a1,b1,c1,d1 are respectively the centroid of the triangles BCD ACD ABD and ABC; AA1, BB1, CC1, DD1 divide one another in the ratio

Answer 1

Answer:

$AB,AC,AD are mutually perpendicular</p><p></p><p>area of ΔABC=∣AB∣∣AC∣=6</p><p></p><p>area of ΔACD=∣AC∣∣AD∣=8</p><p></p><p>area of ΔADB=∣AB∣∣AD∣=10</p><p></p><p>Then the area of △BCD=21∣∣∣∣BC×BD∣∣∣∣</p><p></p><p>area=21∣∣∣∣(AC−AB)×(AD−AB)∣∣∣∣</p><p></p><p>area=21∣∣∣∣AC×AD−AC×AB−AB×AD+AB×AB∣∣∣∣</p><p></p><p>area=21∣∣∣∣AC×AD−AC×AB−AB×AD+0∣∣∣∣</p><p></p><p>area=21∣∣∣∣AC×AD−AC×AB−AB×AD∣∣∣∣</p><p></p><p>area=21∣∣∣∣AC×AD−AC×AB−AB×AD∣∣∣∣</p><p></p><p>area=(2AC×AD)2+(2AC×AB)2+(2AB×AD)2</p><p></p><p>area=(areaofΔACD)2+(areaofΔABC)2+(areaofΔABD)2</p><p></p><p>area=32+42+52</p><p></p><p>area=9+16+25</p><p></p><p>area=50</p><p></p><p>area=52</p><p></p><p></p><p>$