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

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

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 triangles with integer sides and with perimeter 15 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: