Consider a triangle represented by A(0, 0), B(1, 1), C(5, 2). The triangle is…

2014

Consider a triangle represented by A(0, 0), B(1, 1), C(5, 2). The triangle is rotated by 45 degrees about a point P(–1, –1). The co-ordinates of the new triangle obtained after rotation shall be _______ 

  1. A.

    \(A'(-1, \sqrt{2}-1), B'(-1, 2\sqrt{2}-1), C'\left(\frac{3}{2}\sqrt{2}-1, \frac{9}{2}\sqrt{2}-1\right)\)

  2. B.

    \(A'(\sqrt{2}-1, -1), B'(2\sqrt{2}-1, -1), C'\left(\frac{3}{2}\sqrt{2}-1, \frac{9}{2}\sqrt{2}-1\right)\)

  3. C.

    \(A'(-1, \sqrt{2}-1), B'(2\sqrt{2}-1, -1), C'\left(\frac{3}{2}\sqrt{2}-1, \frac{9}{2}\sqrt{2}-1\right)\)

  4. D.

    \(A'(-1, \sqrt{2}-1), B'(2\sqrt{2}-1, -1), C'\left(\frac{9}{2}\sqrt{2}-1, \frac{3}{2}\sqrt{2}-1\right)\)

Attempted by 4 students.

Show answer & explanation

Correct answer: A

Solution: Rotate each vertex about the point (-1,-1) by 45 degrees.

Key facts:

  • Translation to origin (relative to center (-1,-1)): for a point (x,y) use (x+1, y+1).

  • Rotation by 45 degrees uses cos45 = sin45 = √2/2. For a vector (u,v), rotated vector is (u cos45 - v sin45, u sin45 + v cos45).

  • Translate back by adding the center coordinates (-1,-1) (i.e., add -1 to each component of the rotated vector).

Compute each vertex:

  1. For A(0,0): translate → (1,1). Rotate → (1·cos45 - 1·sin45, 1·sin45 + 1·cos45) = (0, √2). Translate back → (-1, √2 - 1).

  2. For B(1,1): translate → (2,2). Rotate → (2·cos45 - 2·sin45, 2·sin45 + 2·cos45) = (0, 2√2). Translate back → (-1, 2√2 - 1).

  3. For C(5,2): translate → (6,3). Rotate → (6·cos45 - 3·sin45, 6·sin45 + 3·cos45) = (3√2/2, 9√2/2). Translate back → (3√2/2 - 1, 9√2/2 - 1).

Final coordinates: A' = (-1, √2 - 1), B' = (-1, 2√2 - 1), C' = (3√2/2 - 1, 9√2/2 - 1).

Explore the full course: Uptet Paper 1