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

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

The number of triangles with integer sides and with perimeter 15 is:

Let $\triangle ABC$ be a triangle right-angled at $B$ with $AB = BC = 18$. The area of the largest rectangle that can be inscribed in this triangle and has $B$ as one of the vertices is:

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

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

Let ABC be an equilateral triangle, with each side of length $k$. If a circle is drawn with diameter AB, then the area of the portion of the triangle lying inside the circle is

In a triangle ABC, let D be the midpoint of BC, and AM be the altitude on BC. If the lengths of AB, BC and CA are in the ratio of 2:4:3, then the ratio of the lengths of BM and AD would be

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:

In a right-angled triangle ABC, the hypotenuse AC is of length 13 cm. A line drawn connecting the midpoints D and E of sides AB and AC is found to be 6 cm in length. The length of BC is

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

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