UPSC CDS Math Area and Perimeter Questions

We have,
The length of hypotenuse =10cm
We know the Pythagoras theorem,
hypotenuse 2= perpendicular 2+ base ‌2
Let, perpendicular is a cm and base is b cm.
Then, 102=a2+b2
⇒b=√100−a2
We know that area =‌

1

2

× perpendicular × base
⇒A=‌

1

2

×a×√100−a2
We want maximum area so we do differentiation of area w.r.t. a which is equal to 0 .
⇒‌

dA

da

=‌

a

4

×‌

(−2a)

√100−a2

+‌

√100−a2

2

=0
⇒‌‌‌

−a2+(100−a2)

2√100−a2

=0
⇒2a2=100
⇒‌‌a=‌

10

√2

cm
‌ Then maximum ‌
a=‌

1

2

×‌

10

√2

×√100−‌

100

2

‌‌=‌

5

√2

×5√2
‌‌=25cm2