Park

In the newly built park will be permanently placed a rotating sprayer irrigation of lawns. Determine the largest radius of the circle which can irrigate by sprayer P so not to spray park visitors on line AB. Distance AB = 55 m, AP = 36 m and BP = 28 m.

Result

r =  16.19 m

Solution:

a=55 b=36 c=28  s=(a+b+c)/2=59.5 r==?  T=ar2=s(sa)(sb)(sc) ar2=s(sa)(sb)(sc) r=2s(sa)(sb)(sc)a r=259.5(59.555)(59.536)(59.528)55 r=16.19 ma = 55 \ \\ b = 36 \ \\ c = 28 \ \\ \ \\ s = (a+b+c)/2 = 59.5 \ \\ r = = ? \ \\ \ \\ T = \dfrac{a r}{2} = \sqrt{s(s-a)(s-b)(s-c)} \ \\ \dfrac{a r}{2} = \sqrt{s(s-a)(s-b)(s-c)} \ \\ r = \dfrac{ 2\sqrt{s(s-a)(s-b)(s-c)}}{a} \ \\ r = \dfrac{ 2 \cdot \sqrt{ 59.5(59.5-55)(59.5-36)(59.5-28)}}{ 55} \ \\ r = 16.19 \ \text{m}

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