If x1 y1 1x2 y2 1x3 y3 1=a1 b1 1a2 b2 1a3 b3 1 then the two triangles with vertices x1,y1,x2,y2,x3,y3 and a1,b1,a2,b2,a3,b3 are

We have x1 y1 1x2 y2 1x3 y3 1=a1 b1 1a2 b2 1a3 b3 1 Hence, the area of triangle with vertices (x1 , y1 ), (x2 , y1 ), (x3 , y3 ), is the same as the area of triangle with vertices (a1 ,b1 ),(a2 ,b2 ),(a3 ,b3 ). Hence, the two triangles are equal in area.

If x1 y1 1x2 y2 1x3 y3 1=a1 b1 1a2 b2 1a3 b3 1 then the two triangles with vertices x1,y1,x2,y2,x3,y3 and a1,b1,a2,b2,a3,b3 are

We have x1 y1 1x2 y2 1x3 y3 1=a1 b1 1a2 b2 1a3 b3 1 Hence, the area of triangle with vertices (x1 , y1 ), (x2 , y1 ), (x3 , y3 ), is the same as the area of triangle with vertices (a1 ,b1 ),(a2 ,b2 ),(a3 ,b3 ). Hence, the two triangles are equal in area.

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If $|\begin{array}{lll} x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1 \end{array}| = |\begin{array}{lll} a_{1} b_{1} 1 \\ a_{2} b_{2} 1 \\ a_{3} b_{3} 1 \end{array}|$ then the two triangles with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ and $(a_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3})$ are

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equal in area

similar

with different areas

congruent

answer is A.

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Detailed Solution

We have $|\begin{array}{lll} x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1 \end{array}| = |\begin{array}{lll} a_{1} b_{1} 1 \\ a_{2} b_{2} 1 \\ a_{3} b_{3} 1 \end{array}|$ Hence, the area of triangle with vertices (x₁ , y₁ ), (x₂ , y₁ ), (x₃ , y₃ ), is the same as the area of triangle with vertices (a₁ ,b₁ ),(a₂ ,b₂ ),(a₃ ,b₃ ). Hence, the two triangles are equal in area.