Symmetry and Transformation：練習問題

Point \(B(-2, 7)\) is reflected in the y-axis to point \(B'\). What are the coordinates of \(B'\)?

Point \(T(-1, 4)\) is rotated clockwise about the origin through \(180^\circ\), then reflected in the \(y\)-axis. Find the final coordinates of the image.

A point P undergoes a sequence of three transformations. First, it is rotated \(180^\circ\) about the origin. Second, the resulting point is reflected across the line \(x = 2\). Finally, the point obtained after reflection is translated by the vector \(\langle 3, -4 \rangle\). If the final position of the point is \((10, -8)\), what were the original coordinates of point P?

A point \(A(3, 5)\) is reflected in the \(x\)-axis to become point \(A'\). What are the coordinates of \(A'\)?

A rhombus \(ABCD\) has vertices at \(A(0, 3)\), \(B(5, 0)\), \(C(0, -3)\), and \(D(-5, 0)\). How many axes of symmetry does this rhombus have?

Point \(P(5, -3)\) is reflected in the line \(y=2\). What are the coordinates of its image \(P'\)?

Point \(B(-2, 3)\) is first translated by the vector \(\begin{pmatrix} 5 \\ -1 \end{pmatrix}\), then its image \(B'\) is reflected across the y-axis. Find the final coordinates of \(B''\).

Point \(K(-1, 5)\) is first reflected in the line \(y=x\), and then translated by the vector \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\). What are the final coordinates of its image \(K''\)?

A triangle has vertices \(X(0, 0)\), \(Y(4, 0)\), and \(Z(0, 3)\).
(a) If the triangle is translated 2 units to the left and 3 units upwards to form triangle \(X'Y'Z'\), find the coordinates of vertex \(Y'\).
(b) State the number of lines of symmetry of the original triangle \(XYZ\).

A triangle has vertices at \(A(2, 1)\), \(B(5, 1)\), and \(C(2, 5)\).

(a) If the triangle is rotated anti-clockwise about the origin through \(90^\circ\) to form triangle \(A'B'C'\), state the coordinates of the image vertex \(B'\).

(b) The image point \(B'\) is further reflected across the \(x\)-axis to form point \(B''\). Find the coordinates of \(B''\).

(c) Does the original triangle \(ABC\) have any lines of reflectional symmetry? Briefly explain your answer based on the lengths of its sides.