Answer:
[tex]\large\boxed{BC=12,\ BA=\dfrac{4}{3}BD}[/tex]
Step-by-step explanation:
Look at the picture.
ΔABC and ΔBDC are similar (AA - If two triangles have two of their angles equal, the triangles are similar).
Therefore the sides are in proportion:
[tex]\dfrac{AC}{BC}=\dfrac{BC}{DC}[/tex]
We have:
AC = 16
BC = x
DC = 9
Substitute:
[tex]\dfrac{16}{x}=\dfrac{x}{9}[/tex] cross multiply
[tex]x^2=(16)(9)\to x=\sqrt{(16)(9)}\\\\x=\sqrt{16}\cdot\sqrt9\\\\x=4\cdot3\\\\x=12[/tex]
Therefore BC = 12.
Calculate the similarity scale:
[tex]k=\dfrac{AC}{BC}\to k=\dfrac{16}{12}=\dfrac{16:4}{12:4}=\dfrac{4}{3}[/tex]
Therefore we have the poportion:
[tex]\dfrac{BA}{BD}=\dfrac{4}{3}[/tex] cross multiply
[tex]3BA=4BD[/tex] divide both sides by 3
[tex]BA=\dfrac{4}{3}BD[/tex]
