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(A) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Given midpoints are $(5, 0)$,$(5, 12)$,and $(0, 12)$.Using the m

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$(0, 0)$

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(A) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Given midpoints are $(5, 0)$,$(5, 12)$,and $(0, 12)$.Using the midpoint formula:$\frac{x_1 + x_2}{2} = 5, \frac{y_1 + y_2}{2} = 0$$\frac{x_2 + x_3}{2} = 5, \frac{y_2 + y_3}{2} = 12$$\frac{x_3 + x_1}{2} = 0, \frac{y_3 + y_1}{2} = 12$Solving these equations:Summing $x$-coordinates: $x_1 + x_2 + x_3 = 10$. Thus,$x_3 = 0, x_1 = 0, x_2 = 10$.Summing $y$-coordinates: $y_1 + y_2 + y_3 = 24$. Thus,$y_3 = 24, y_1 = 0, y_2 = 0$.The vertices are $A(0, 0)$,$B(10, 0)$,and $C(0, 24)$.Since the triangle is a right-angled triangle with the right angle at the origin $(0, 0)$,the orthocenter is the vertex at the right angle,which is $(0, 0)$.

If the midpoints of the sides of a triangle are $(5, 0)$,$(5, 12)$,and $(0, 12)$,then what is the orthocenter of this triangle?

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