Question

There are 3 villages A,B and C such that the distance from A to B is 7Km, from B to C is 5 Km and from C to A is 8 Km. The gram pradhan wants to dig a well in such a way that the distance from each villages are equal. What should be the location of well?

Answer:

The distance from the villages to the well is [7√3]/3 km.

ABC forms a triangle with AB = 7km, BC = 5km and AC = 8km. The circumcenter of a triangle is equidistant from all its vertices as the three vertices lie on the circumcircle and so the distance from the circumcenter to any one of the vertices is its radius.
Semiperimeter of the triangle so formed = 8+7+5/2 = 20/2 = 10 km. Area of the triangle = sqrt[10(10-8)(10-7)(10-5)] = sqrt(10*2*3*5) = 10√3 sq km As we know,
Radius of the circumcircle = Product of the lengths of three sides/4(Area of triangle)
we have the distance from the villages to the well
= (8*7*5)/[4*10√3] = 7/√3 = 7√3/3 km


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