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Let A(3, 4) , B(5 , 5 ) and C(5 , -5 ) be the vertices of a triangle, where (0, 4 ) . If G(h, 1) and H(a, b) are the centroid and orthocenter of the triangle respectively, then the value of (a + b + 25 2 ) is equal to ______ .

Correct answer

37

Step-by-step solution

The vertices of the triangle are A(3, 4) , B(5 , 5 ) and C(5 , -5 ) . The distance of each vertex from the origin (0, 0) is: OA = 3^2 + 4^2 = 5 OB = (5 )^2 + (5 )^2 = 5 OC = (5 )^2 + (-5 )^2 = 5 Since OA = OB = OC , the circumcenter O of the triangle is the origin (0, 0) . The centroid G(h, k) has coordinates: h = 3 + 5 + 5 3 k = 4 + 5 - 5 3 Given that the y -coordinate of the centroid is 1 , we have: 4 + 5( - ) 3 = 1 5( - ) = -1 Squaring both sides: 25( ^2 + ^2 - 2 ) = 1 25(1 - 2 ) = 1 25 2 = 24 Using the algebrai

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