UPSC CDS 1 2022 Math Paper

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