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If the square \(A B C D\) where \(A(0,0), B(2,0), C(2,2)\) and \(D(0,2)\) undergoes the following three transformations successively (i) \(f_{1}(x, y) \rightarrow(y, x)\) (ii) \(f_{2}(x, y) \rightarrow(x+3 y, y)\) (iii) \(f_{3}(x, y) \rightarrow\left(\frac{x-y}{2}, \frac{x+y}{2}\right)\) then the final figure is a : (a) square (b) parallelogram (c) rhombus (d) none of these

Short Answer

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The final figure after applying all three transformations is none of the provided options.
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Step 1: Apply first transformation \(f_{1}(x, y) \rightarrow(y, x)\)

Replacing each point \((x, y)\) with \((y, x)\) gives: \(A'(0,0), B'(0,2), C'(2,2)\) and \(D'(2,0)\). Note that this transformation interchange x and y coordinates of each point.

Step 2: Apply second transformation \(f_{2}(x, y) \rightarrow(x+3 y, y)\)

Applying the second transformation to each point obtained from first step, we get: \(A''(0, 0), B''(6, 2), C''(8, 2)\) and \(D''(2, 0)\). Note that for each point, the x-coordinate is replaced by the sum of x-coordinate and 3 times the y-coordinate, and the y-coordinate remains unchanged.

Step 3: Apply third transformation \(f_{3}(x, y) \rightarrow\left(\frac{x-y}{2}, \frac{x+y}{2}\right)\)

By applying this transformation to each point achieved from second step, we get new points as: \(A'''(0,0), B'''(2,4), C'''(3,5)\) and \(D'''(-1,1)\). This transformation replaces each x-coordinates with the average of the x and y-coordinates and y-coordinate with the half of the sum of the x and y coordinates.

Step 4: Identify the shape

After all the transformations, the coordinates of the resulting shape are \(A'''(0,0), B'''(2,4), C'''(3,5)\), and \(D'''(-1,1)\). If we calculate the distances of consecutive points, we notice that the opposite sides are not equal, therefore, this is not a square or a rhombus. Also, the opposite sides are not parallel which rules out the possibility of a parallelogram. Hence, the resulting figure does not fall into any of the provided categories, so the answer is (d) none of these.

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