Congruence and similarity：練習問題

In the diagram, triangles \(ABC\) and \(ADE\) are similar. If \(BC\) is parallel to \(DE\), \(AB = 4\text{ cm}\), \(BD = 2\text{ cm}\), and the area of triangle \(ABC\) is \(12\text{ cm}^2\), find the area of triangle \(ADE\).

Two similar triangles have corresponding sides in the ratio \( 3 : 5 \). If the area of the smaller triangle is \( 18\text{ cm}^2 \), find the area of the larger triangle.

In triangle \(ABC\), \(X\) and \(Y\) are points on \(AB\) and \(AC\) respectively such that \(AX = \frac{1}{3} AB\) and \(AY = \frac{1}{3} AC\). If the area of triangle \(ABC\) is \(k\), express the area of triangle \(AXY\) in terms of \(k\).

A map is drawn to a scale of \(1 : 20\,000\). A forest is represented by an area of \(4.5\text{ cm}^2\) on the map. Calculate the actual area of the forest in square kilometres.

In triangle \(ABC\), point \(D\) lies on \(AB\) and point \(E\) lies on \(AC\) such that \(DE\) is parallel to \(BC\). If \(AD = 4\text{ cm}\), \(DB = 6\text{ cm}\), and the area of triangle \(ADE\) is \(8\text{ cm}^2\), calculate the area of the trapezium \(DBCE\).

Two similar containers have capacities of \(250\text{ ml}\) and \(2000\text{ ml}\) respectively. If the height of the smaller container is \(6\text{ cm}\), find the height of the larger container.

Triangle \(ABC\) is similar to triangle \(PQR\). The lengths of \(AB\) and \(PQ\) are \(4\text{ cm}\) and \(10\text{ cm}\) respectively. If the perimeter of triangle \(ABC\) is \(15\text{ cm}\) and the difference between the areas of the two triangles is \(84\text{ cm}^2\), find the area of triangle \(PQR\).

Two mathematically similar solid pyramids have vertical heights in the ratio \(3 : 5\). The larger pyramid is made of a material with a density of \(4\text{ g/cm}^3\) and has a mass of \(3750\text{ g}\). Given that the smaller pyramid is made of a material with a density of \(2\text{ g/cm}^3\), find the mass of the smaller pyramid.

Triangle \(ABC\) and triangle \(PQR\) are mathematically similar. The ratio of the length of \(AB\) to the length of \(PQ\) is \(2 : 3\).
(a) Given that the area of triangle \(ABC\) is \(24\text{ cm}^2\), find the area of triangle \(PQR\).
(b) If the perimeter of triangle \(PQR\) is \(45\text{ cm}\), find the perimeter of triangle \(ABC\).

In triangle \(PQR\), \(X\) is a point on \(PQ\) and \(Y\) is a point on \(PR\) such that \(XY\) is parallel to \(QR\). The line \(QY\) intersects \(RX\) at point \(O\).
(a) State the ratio of \(PX\) to \(PQ\) given that \(PX : XQ = 2 : 3\).
(b) Find the ratio of the area of triangle \(PXY\) to the area of triangle \(PQR\).
(c) If the area of triangle \(OXQ\) is \(27\text{ cm}^2\), find the area of triangle \(OXY\).