ΔABD and ΔACD are similar (By A-A-A) and The length of DA is 6.
The triangles where all the corresponding sides of the two triangles are in equal proportion are called similar triangles.
In triangle ΔABC ∠BAC=90°
Let ∠ABC=x
then ∠ACB=90°-x
In triangle ΔADC, ∠ADC =90°
∠ACB=∠ACD=90°-x
∠DAC= 180°-(∠ADC+∠ACD)= 180°-(90°+90°-x)= x
In triangle ΔADB , ∠ADB =90°
∠ABD=x
∠BAD=90°-x
Between triangles ΔABD and ΔACD
∠ABD=∠DAC (=X)
∠BAD∠ACD (=90°-x)
∠ADB=∠ADC (=90°)
from the above, it is clear that triangles ΔABD nad ΔACD is similar. (By A-A-A)
So sides are in equal proportion in 2 triangles,
AD/DC= BD/AD
⇒AD²= BD*DC
⇒AD²=9*4
⇒AD²=36
⇒AD=6
⇒DA=6
Therefore triangles ΔABD and ΔACD are similar (By A-A-A) and The length of DA is 6.
Learn more about similar triangle
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