Consider a triangle A(0,0), B(1, 1) and C(5, 2). The triangle has to be…
2015
Consider a triangle A(0,0), B(1, 1) and C(5, 2). The triangle has to be rotated by an angle of 45𝑜 about the point P(-1, -1). What shall be the coordinates of new triangle ?
- A.
\(A' = (1, \sqrt{2}-1), B' = (-1, 2\sqrt{2}-1) \text{ and } C'=\bigl(3 \sqrt{2} -1, \frac{9}{2} \sqrt{2}-1 \bigr)\) - B.
\(A' = (1, \sqrt{2}-1), B' = (2\sqrt{2}-1, -1) \text{ and } C'=\bigl(3 \sqrt{2} -1, \frac{9}{2} \sqrt{2}-1 \bigr)\) - C.
\(A' = (-1, \sqrt{2}-1), B' = (-1, 2\sqrt{2}-1) \text{ and } C'=\bigl(3 \sqrt{2} -1, \frac{9}{2} \sqrt{2}-1 \bigr)\) - D.
\(A' = (\sqrt{2}-1,-1) B' = (-1, 2\sqrt{2}-1) \text{ and } C'=\bigl(3 \sqrt{2} -1, \frac{9}{2} \sqrt{2}-1 \bigr)\)
Attempted by 62 students.
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Correct answer: C
Solution outline:
Translate each point so the pivot P(-1,-1) moves to the origin by adding (+1, +1) to coordinates.
Rotate the translated point by 45°. With cos45° = sin45° = √2/2, the rotated coordinates (relative to the pivot) are:
x_rot = ((x + 1) - (y + 1)) × √2/2
y_rot = ((x + 1) + (y + 1)) × √2/2
Translate back by subtracting (1, 1) to get final coordinates: x' = x_rot - 1, y' = y_rot - 1.
Apply the steps to each vertex:
For A(0,0): translate → (1,1). Rotate: x_rot = (1-1)×√2/2 = 0, y_rot = (1+1)×√2/2 = √2. Translate back → A' = (-1, √2 - 1).
For B(1,1): translate → (2,2). Rotate: x_rot = (2-2)×√2/2 = 0, y_rot = (2+2)×√2/2 = 2√2. Translate back → B' = (-1, 2√2 - 1).
For C(5,2): translate → (6,3). Rotate: x_rot = (6-3)×√2/2 = 3√2/2, y_rot = (6+3)×√2/2 = 9√2/2. Translate back → C' = (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).