A circular sprinkler is placed inside a triangular garden with sides 13 m, 14 m, and 15 m. What is the radius of the sprinkler if it touches all three sides of the triangle?
4 m
A circle that touches all three sides of a triangle is its incircle, and its radius (the inradius) is given by \(r = \frac{\text{Area}}{s}\), where \(s\) is the semi-perimeter.
Semi-perimeter \(s = \frac{13+14+15}{2} = \frac{42}{2} = 21 \text{ m}\).
Using Heron's formula, Area \(= \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21(21-13)(21-14)(21-15)}\).
\(= \sqrt{21\times 8\times 7\times 6} = \sqrt{7056} = 84 \text{ m}^2\).
Inradius \(r = \frac{\text{Area}}{s} = \frac{84}{21} = 4 \text{ m}\).
Key concept: the inradius equals the triangle's area divided by its semi-perimeter, with Heron's formula supplying the area from the three sides.
Hence the radius of the sprinkler is 4 m.
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