UPSC CDS Math Area and Perimeter Questions

Let the shorter sides be x and y, respectively, and x > y.
By Pythagoras theorem:
x2+y2=102....(1)
Also, area of triangle:

1

2

xy=24⇒xy=48
Putting y=

48

x

in equation (1), we get:
x2+(

48

x

)2=102
⇒x4+2304=100x2
⇒x4−100x2+2304=0
Let t=x2
⇒t2−100t+2304=0
⇒t=36,64
∴x2=36 or 64
⇒x=6 or 8
y=8 or 6
Since x > y, therefore, x = 8 cm and y = 6 cm
When base is halved, we get new base of triangle = 16 cm
When height is doubled, we get new height = 3 cm
New hypotenuse =√162+32
=√265cm