The perimeter of a triangle is 40 cm and its area is 60 sq. cm. If the largest side measures 17 cm, then the length (in cm) of the smallest side of the triangle is:

A. 4 A. 4

B. 6 B. 6

C. 8 C. 8

D. 15 D. 15

Given:
- Perimeter of triangle = 40 cm
- Area of triangle = 60 sq. cm
- Largest side = 17 cm
Let the sides of the triangle be: a ≤ b ≤ c = 17
Perimeter: a + b + c = 40
a + b + 17 = 40
a + b = 23
Use Heron's formula for area:
s = perimeter / 2 = 40 / 2 = 20
Area = √[s(s − a)(s − b)(s − c)] = 60
Substitute s and c:
60 = √[20 × (20 − a) × (20 − b) × (20 − 17)]
60 = √[20 × (20 − a) × (20 − b) × 3]
60² = 60 × (20 − a) × (20 − b)
3600 = 60 × (20 − a) × (20 − b)
(20 − a)(20 − b) = 60
Since a + b = 23 → b = 23 − a:
(20 − a)(20 − (23 − a)) = 60
(20 − a)(−3 + a) = 60
20a − a² − 60 + 3a = 60
−a² + 23a − 60 − 60 = 0 → −a² + 23a − 120 = 0
Multiply by −1: a² − 23a + 120 = 0
Factorize: (a − 8)(a − 15) = 0 → a = 8 or 15
Smallest side: a = 8 cm
Correct Answer: C. 8

দেওয়া আছে:
- ত্রিভুজের পরিধি = 40 cm
- ক্ষেত্রফল = 60 sq. cm
- সর্বোচ্চ বাহু = 17 cm
ধরি ত্রিভুজের বাহুগুলো: a ≤ b ≤ c = 17
পরিধি: a + b + c = 40
a + b + 17 = 40
a + b = 23
Heron সূত্র ব্যবহার করি:
s = perimeter / 2 = 40 / 2 = 20
ক্ষেত্রফল = √[s(s − a)(s − b)(s − c)] = 60
s এবং c বসাই:
60 = √[20 × (20 − a) × (20 − b) × (20 − 17)]
60 = √[20 × (20 − a) × (20 − b) × 3]
60² = 60 × (20 − a) × (20 − b)
3600 = 60 × (20 − a) × (20 − b)
(20 − a)(20 − b) = 60
a + b = 23 → b = 23 − a:
(20 − a)(20 − (23 − a)) = 60
(20 − a)(−3 + a) = 60
20a − a² − 60 + 3a = 60
−a² + 23a − 120 = 0
a² − 23a + 120 = 0
গুণফল: (a − 8)(a − 15) = 0 → a = 8 বা 15
ক্ষুদ্রতম বাহু: a = 8 cm
সঠিক উত্তর: C. 8

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