 Suggested languages for you:

Europe

Problem 25

# 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

Expert verified
The final figure after applying all three transformations is none of the provided options.
See the step by step solution

## 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.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features ## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner 