$\mathrm{ABCD}$is a quadrilateral. The diagonals of $\mathrm{ABCD}$intersect at the point $\mathrm{P}$. The area of the triangles $\mathrm{APD}$and $\mathrm{BPC}$are $27$and $12$, respectively. If the areas of the triangles $\mathrm{APB}$and $\mathrm{CPD}$are equal then the area of triangle $\mathrm{APB}$is

In a right angle triangle $\mathrm{BAC}$given below, $\text{AD }$is the altitude of the hypotenuse $\mathrm{BC}$. The figure is followed by three possible inferences.

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$1$. Triangle $\mathrm{ABD}$and triangle $\mathrm{CAD}$are similar

$2$. Triangle $\mathrm{ADB}$and triangle $\mathrm{CDA}$are congruent.

$3$. Triangle $\mathrm{ADB}$and triangle $\mathrm{CAB}$are similar.

In KyaKya Island, there is a circular park. There are four points of entry into the park, namely –$\mathrm{P}. \mathrm{Q}. \mathrm{R} \mathrm{and} \mathrm{S}.$ The king of the island His excellency Mr. Honolulu got three paths constructed which connected the points $\mathrm{PQ}, \mathrm{RS}, \mathrm{and} \mathrm{PS}$. The length of the path $\mathrm{PQ}$is $10 \mathrm{units}$, and the length of the path RSIs $7 \mathrm{units}.$ Later, the municipal corporation extended the paths $\mathrm{PQ}$and $\mathrm{RS}$past $\mathrm{Q}$and $\mathrm{R}$respectively, and they meet at a point $\mathrm{T}$on the main road outside the park. The path from $\mathrm{Q}$to $\mathrm{T}$measures $8$units, and it was found that the $\mathrm{ZPTS}$ is $60°.$ Find the area (in square units) enclosed by the paths $\mathrm{PT}, \mathrm{TS}, \mathrm{and} \mathrm{PS}.$

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