A rectangular sheet of paper, when halved by folding it at the midpoint of its longer side, results in a rectangle, whose longer and shorter sides are in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is $2,$ what is the area of the smaller rectangle?

On a semicircle with diameter $\mathrm{AD},$ chord $\mathrm{BC}$ is parallel to the diameter. Further, each of the chords $\mathrm{AB}$ and $\mathrm{CD}$ has length $2,$ while $\mathrm{AD}$ has length $8.$ What is the length of $\mathrm{BC} ?$

Two identical circles intersect so that their centers, and the points at which they intersect, form a square of side $1 \mathrm{cm}.$ The area in $\mathrm{sq}. \mathrm{cm}$ of the portion that is common to the two circles is

A jogging park has two identical circular tracks touching each other, and a rectangular track enclosing the two circles. The edges of the rectangles are tangential to the circles. Two friends, $\mathrm{A}$ and $\mathrm{B},$ start jogging simultaneously from the point where one of the circular tracks touches the smaller side of the rectangular track. $\mathrm{A}$ jogs along the rectangular track, while $\mathrm{B}$ jogs along the two circular tracks in a figure of eight. Approximately, how much faster than $\mathrm{A}$ does $\mathrm{B}$ have to run, so that they take the same time to return to their starting point ?

Consider the triangle $\mathrm{ABC}$ shown in the following figure where $\mathrm{BC}=12 \mathrm{cm}, \mathrm{DB}=9 \mathrm{cm}, \mathrm{CD}=6 \mathrm{cm}$ and $\angle \mathrm{RCD}=\angle \mathrm{RAC}$

What is the ratio of the perimeter of $∆\mathrm{ADC}$ to that of the $∆\mathrm{BDC} ?$