Transformations：練習問題

A point \(A(2, 3)\) is translated 4 units to the left and 2 units up to point \(B\). Then, point \(B\) is rotated \(180^{\circ}\) about the origin to point \(C\). Find the coordinates of point \(C\).

The point \(P(4, -2)\) is reflected in the \(y\)-axis to point \(Q\). Point \(Q\) is then translated 3 units upwards to point \(R\). Find the coordinates of \(R\).

A point \(A(4, 7)\) is translated 5 units to the left and 2 units downwards to a new position \(A'\). Find the coordinates of \(A'\).

A point \(P(3, -5)\) is reflected across the \(x\)-axis to point \(Q\), and then \(Q\) is rotated counter-clockwise about the origin by \(90^{\circ}\) to point \(R\). Find the coordinates of point \(R\).

A triangle \(T\) has a vertex at \(G(a, b)\). It is first reflected across the \(y\)-axis, then rotated \(270^{\circ}\) clockwise about the origin, and finally translated 4 units up. If the final position of the vertex is \((5, 2)\), find the values of \(a\) and \(b\).

A point \(A(4, 2)\) is translated 3 units to the left and 5 units downwards to point \(B\).
(a) Find the coordinates of point \(B\).
(b) Point \(B\) is then reflected about the \(y\)-axis to point \(C\). Find the coordinates of point \(C\).

In a Cartesian plane, a triangle has vertices at \(X(2, 1)\), \(Y(5, 1)\), and \(Z(2, 4)\).
(a) The triangle is reflected about the \(x\)-axis to form triangle \(X'Y'Z'\). State the coordinates of \(X'\), \(Y'\), and \(Z'\).
(b) Triangle \(X'Y'Z'\) is then translated 4 units to the right and 2 units upwards to form triangle \(X''Y''Z''\). Find the coordinates of \(X''\), \(Y''\), and \(Z''\).
(c) Calculate the area of triangle \(XYZ\).