UPSC CDS I 2025 Mathematics Paper

Perimeter of triangle ABC=105cm
Ratio of sides AB:BC:CA=5:10:6
Formula used:
Heron's Formula for the area of a triangle:
Area =(√s(s−a)(s−b)(s−c)
Where a,b,c are the lengths of the sides of the triangle, and
s is the semi-perimeter s=(‌

a+b+c

2

)
Calculations:
From the previous problem, we established
The ratio of the sides AB:BC:CA=5:10:6.
Let the sides be AB=5x,BC=10x, and CA=6x.
The perimeter is the sum of the sides:
Perimeter =AB+BC+CA=5x+10x+6x=21x
Given Perimeter =105cm.
⇒21x=105
⇒x=(‌

105

21

)
⇒x=5
a(BC)=10x=10×5=50cm
b(CA)=6x=6×5=30cm
c(AB)=5x=5×5=25cm
The semi-perimeter (s):s=(‌

‌ Perimeter ‌

2

)=(‌

105

2

)=52.5cm
Area=(√s(s−a)(s−b)(s−c)

‌‌ Area ‌=(√‌52.5×(52.5−50)×(52.5−30)×(52.5−25))
‌‌ Area ‌=(√‌52.5×2.5×22.5×27.5)
‌‌ Area ‌=(√‌99738.28125)
‌‌ Area ‌≈284.975cm2
‌‌ Area ‌≈285cm2