Similarity：練習問題

In the diagram, triangle \(ABC\) is similar to triangle \(XYZ\). Side \(AB\) corresponds to side \(XY\). Given that \(AB = 3\text{ cm}\), \(AC = 6\text{ cm}\) and \(XY = 12\text{ cm}\), calculate the length of side \(XZ\).

Two mathematically similar rectangles have a linear scale factor of 1 : 3. If the area of the smaller rectangle is \( 15\text{ cm}^2 \), calculate the area of the larger rectangle.

In triangle \(PQR\), a line segment \(ST\) is drawn parallel to \(QR\), where \(S\) lies on \(PQ\) and \(T\) lies on \(PR\). If the lengths are \(PS = x\), \(SQ = x + 2\), \(ST = 6\), and \(QR = 15\), calculate the value of \(x\).

Two mathematically similar pentagons have corresponding sides in the ratio \( 2 : 3 \). Given that the area of the smaller pentagon is \( 24\text{ cm}^2 \), calculate the area of the larger pentagon.

Two mathematically similar solid spheres have volumes of \(54\text{ cm}^3\) and \(128\text{ cm}^3\). If the surface area of the smaller sphere is \(45\text{ cm}^2\), calculate the surface area of the larger sphere.

Two geometrically similar solids have corresponding lengths in the ratio $$2:3$$. If the volume of the smaller solid is $$24 \text{ cm}^3$$, calculate the volume of the larger solid.