Concept: Criteria for Similarity of Triangles: Angle-Angle-Angle: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. Pythagoras theorem: In a right angle triangle Hypotenuse 2= Base 2+ Perpendicular 2 Calculation:
In an △ABC&△ABD ∠A=∠A= common ∠ABC=∠BDA=90∘ Hence, △ABC is similar to △ABD . Therefore, both triangles have the same ratio of corresponding sides ⇒
AD
AB
=
AB
AC
⇒AB2=AD×AC ⇒AB2=4×9=36 ⇒AB=6 unit In and △ABC, using Pythagoras theorem ⇒AC2=AB2+BC2 ⇒81=36+BC2 ⇒BC=√45=3√5 ⇒AB+BC=6+3√5 ⇒AB+BC=6+3×2.23 ∴AB+BC≈12.6