Let $\triangle ABC$ be a triangle with $AB = AC$ and $D$ be a point on $BC$ such that $\angle BAD = 30^\circ$. If $E$ is a point on $AC$ such that $AD = AE$, then $\angle CDE$ equals

The side AB of a triangle ABC is c. The median BD is of length k. If $\angle BDA = \theta$ and $\theta < 90^\circ$, then the area of triangle ABC is

In triangle $ABC, AB = AC = x, ∠ABC = \theta$ and the circumradius is equal to $y$. Then $\frac{x}{y}$ equals

The number of points, having both coordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 31),$ and $(31, 0)$ is

The number of acute angled triangles whose sides are three consecutive positive integers and whose perimeter is at most 100 is

The lengths of the sides of a triangle are $x, 21$ and $40$, where $x$ is the shortest side. A possible value of $x$ is:

Consider a triangle with side lengths 4 meters, 6 meters, and 9 meters. A dog runs around the triangle in such a way that the shortest distance of the dog from the triangle is exactly 1 meter. The total distance covered (in meters) by the dog in one round is