The triangle O A B has O = ( 0 , 0 ) , A = ( 2 , 0 ) , B = ( 1 , 3 ) . An interior point P satisfies d ( P , O A ) + d ( P , A B ) + d ( P , OB ) = 3 . If the area of the region of all such P is k

Question

The triangle O A B has O = ( 0 , 0 ) , A = ( 2 , 0 ) , B = ( 1 , 3 ​ ) . An interior point P satisfies d ( P , O A ) + d ( P , A B ) + d ( P , OB ) = 3 ​ . If the area of the region of all such P is k , find k 3 ​ .

TestHub · Accepted Answer

The triangle is equilateral, with side 2, area 3 ​ , and altitude 3 ​ . Summing the areas of △ PO A , △ P A B , and △ POB shows that the sum of distances to its sides is the altitude for every interior point. Thus k = 3 ​ and k 3 ​ = 3 .

Mathematics - Properties of Triangle Question with Solution | TestHub

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