Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $2 m \angle P = 3 m \angle Q$ and $m \angle C = 100^{\circ}$,find $m \angle B$. (in $^{\circ}$)

Prove that the area of the equilateral triangle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

In rectangle $ABCD$,$AB = 9$ and $BC = 12$. Then,$BD = \ldots$

The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}$. Is it correct to say that the ratio of their areas is $\frac{6}{5}$? Why?

In $\Delta ABC$,$A-D-B$,$A-E-C$ and $\overline{DE} \parallel \overline{BC}$. If $AB:AC = 3:4$,which of the following holds true?

If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,then $BC^2 : YZ^2 = \ldots \ldots \ldots$ (Wait,the question asks for $ABC : XYZ = BC^2 : \ldots$ which refers to the ratio of areas). Given $\Delta ABC \sim \Delta XZY$,the ratio of their areas is equal to the ratio of the squares of their corresponding sides. Thus,$\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta XZY)} = \frac{BC^2}{ZY^2}$.