Congruence and Similarity: แบบฝึกหัด

Which of the following describes the condition for two triangles to be similar if the ratios of their three pairs of corresponding sides are equal?

In \(\triangle PQR\), \(\angle P = 45^\circ\) and \(\angle Q = 75^\circ\). In \(\triangle STU\), \(\angle S = 45^\circ\) and \(\angle U = 60^\circ\). Are the two triangles similar?

In the figure, \(\triangle ABC\) is a triangle where \(\angle BAC = 90^\circ\) and \(AD \perp BC\) at \(D\). If \(AB = 15 \text{ cm}\) and \(BC = 25 \text{ cm}\), find the length of \(BD\).

If two triangles are congruent, what is the ratio of their perimeters?

Consider two similar solid cylinders, \(S_1\) and \(S_2\). The ratio of the surface area of \(S_1\) to the surface area of \(S_2\) is \(k^2:1\). If the volume of \(S_1\) is \(108\pi\) and the volume of \(S_2\) is \(32\pi\), find the value of \(k\).

Two triangles, \(\triangle XYZ\) and \(\triangle LMN\), are congruent. If \(\angle X = 40^\circ\) and \(\angle Y = 60^\circ\), what is the measure of \(\angle M\)?

Two similar spheres have a ratio of surface areas \(A_1 : A_2 = 16 : 49\). If the volume of the smaller sphere (\(V_1\)) is \(128 \pi\) cm\({^3}\), find the volume of the larger sphere (\(V_2\)).

In the right-angled triangle \(ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\) at point \(D\). If \(AD = 4\) cm and \(CD = 9\) cm, find the length of \(BD\) by using the similarity of triangles.

In the figure, \(\triangle ABC\) and \(\triangle DEF\) are two triangles. It is given that \(AB = DE\), \(BC = EF\), and \(\angle B = \angle E\).

(a) State the congruence condition that proves \(\triangle ABC \cong \triangle DEF\).

(b) If \(AB = 5\text{ cm}\), \(BC = 12\text{ cm}\), and \(\angle B = 90^\circ\), find the length of \(DF\).

(c) If the area of \(\triangle ABC\) is \(30\text{ cm}^2\), find the area of \(\triangle DEF\).

In \(\triangle ABC\), \(\angle BAC = 90^\circ\). \(AD\) is the altitude drawn from point \(A\) to the hypotenuse \(BC\), where \(D\) lies on \(BC\). It is given that \(BD = 4\) cm and \(CD = 9\) cm.

(a) Show that \(\triangle ABD \sim \triangle CAD\).
(b) Find the length of \(AD\).
(c) Find the ratio of the Area of \(\triangle ABD\) : Area of \(\triangle CAD\).