Answer: B
Triangles ABD and ABC are similar. ∴ \(\cfrac{BD}{AB} = \cfrac{AB}{BC}\) i.e., \(AB^2 = BD \times BC\) — (i) Again, triangles ADC and ABC are similar. ∴ \(\cfrac{CD}{AC} = \cfrac{AC}{BC}\) i.e., \(AC^2 = CD \times BC\) — (ii) From (i) and (ii), we get: \(\cfrac{BD \times BC}{CD \times BC} = \cfrac{AB^2}{AC^2}\) i.e., \(\cfrac{BD}{DC} = \left(\cfrac{AB}{AC}\right)^2\) ⇒ \(\cfrac{BD}{DC} = \cfrac{9}{16}\) ∴ BD : DC = 9 : 16
Triangles ABD and ABC are similar. ∴ \(\cfrac{BD}{AB} = \cfrac{AB}{BC}\) i.e., \(AB^2 = BD \times BC\) — (i) Again, triangles ADC and ABC are similar. ∴ \(\cfrac{CD}{AC} = \cfrac{AC}{BC}\) i.e., \(AC^2 = CD \times BC\) — (ii) From (i) and (ii), we get: \(\cfrac{BD \times BC}{CD \times BC} = \cfrac{AB^2}{AC^2}\) i.e., \(\cfrac{BD}{DC} = \left(\cfrac{AB}{AC}\right)^2\) ⇒ \(\cfrac{BD}{DC} = \cfrac{9}{16}\) ∴ BD : DC = 9 : 16