Shortcut Trick
Identify the Pythagorean triplet: The sides 220, 231, and 319 are in the ratio 20 : 21 : 29 (divided by 11).
Since 202 + 212 = 292, the field is a right-angled triangle.
Area of triangle = 1/2 × 220 × 231 = 25410 m2.
Area of circular garden = 25410 ÷ 15 = 1694 m2.
πr2 = 1694 ⇒ (22/7) × r2 = 1694 ⇒ r2 = 77 × 7 = 539.
r = √539 = 7√11 m ⇒ Diameter = 2 × 7√11 = 14√11 m.
∴ The correct answer is 14√11 m.
Alternate Method
Given:
Sides of the triangular field: a = 220 m, b = 231 m, c = 319 m.
Relation: Area of triangle = 15 × Area of circular garden.
Formula Used:
Area of triangle (Right-angled) = 1/2 × base × height
Area of circle = πr2
Diameter = 2r
Calculations:
⇒ Checking for right-angled triangle: 2202 + 2312 = 48400 + 53361 = 101761
⇒ 3192 = 101761
⇒ Since a2 + b2 = c2, the triangle is right-angled at C.
⇒ Area of triangular field = 1/2 × 220 × 231 = 110 × 231 = 25410 m2
⇒ Area of circular garden = 25410 ÷ 15 = 1694 m2
⇒ πr2 = 1694
⇒ (22/7) × r2 = 1694
⇒ r2 = (1694 × 7) ÷ 22
⇒ r2 = 77 × 7 = 539
⇒ r = √539 = √(49 × 11) = 7√11 m
⇒ Diameter = 2r = 2 × 7√11 = 14√11 m
∴ The correct answer is 14√11 m.
Additional Information
Pythagorean Triplets
A set of three positive integers a, b, and c such that a2 + b2 = c2. Common examples include (3, 4, 5), (5, 12, 13), and (20, 21, 29).
Heron's Formula
For any triangle with sides a, b, c, Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter.
Circle Properties
Area = πr2 and Circumference = 2πr. The diameter is the longest chord and is equal to twice the radius (d = 2r).