Let $A B C$ be a right-angled triangle with $B C$ as the hypotenuse. Lengths of $A B$ and AC are $15 \mathrm{~km}$ and $20 \mathrm{krn}$, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of $30 \mathrm{~km}$ per hour is

Suppose the medians $BD$ and $CE$ of a triangle $ABC$ intersect at a point $O$. If area of triangle $ABC$ is $108$ sq. cm., then, the area of the triangle $EOD$, in sq. cm., is

From a triangle $ABC$ with sides of lengths $40$ ft, $25$ ft and $35$ ft, a triangular portion $GBC$ is cut off where $G$ is the centroid of $ABC$. The area, in sq ft, of the remaining portion of triangle $ABC$ is

The midpoints of sides $AB$, $BC$, and $AC$ in $\triangle ABC$ are $M$, $N$, and $P$, respectively. The medians drawn from $A$, $B$, and $C$ intersect the line segments $MP$, $MN$ and $NP$ at $X$, $Y$, and $Z$, respectively. If the area of $\triangle ABC$ is $1440$ sq cm, then the area, in sq cm, of $\triangle XYZ$ is

The sum of the perimeters of an equlateral triangle and a rectangle is $90 \mathrm{~cm}$ the area, $T$ , of the triangle and the area, $R$ , of the rectangle, both in sq cm, satisfy the relationship $R=T^{2}$. If the sides of the rectangle are in the ratio $1: 3$, then the length, in cm , of the longer side of the rectangle, is

Given an equilateral triangle $\mathrm{T} 1$ with side $24 \mathrm{~cm}$, a second triangle $\mathrm{T} 2$ is formed by joining the midpoints of the sides of $\mathrm{T} 1$. Then a third triangle $\mathrm{T} 3$ is formed by joining the midpoints of the sides of $\mathrm{T} 2$. If this process of forming triangles is continued, the sum of the areas, in sq cm , of infinitely many such triangles $\mathrm{T} 1, \mathrm{~T} 2, \mathrm{~T} 3, \ldots$ will be

From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the three perpendiculars is $s$. Then the area of triangle is