O P Malhotra, S K Gupta and, Anubhuti Gangal Solutions for Exercise 1: Exercise

The line joining the midpoints of two sides of a triangle is _____ to the third side.

The line drawn through the midpoint of one side of a triangle parallel to another side _____ the third side.

In the quadrilateral $ABCD$, the midpoints of $AB, BC, CD, DA$ are $L, M, P, Q$. Using the midpoint theorem make a statement concerning the lengths and directions of $LM$ and $AC$.

In the quadrilateral $ABCD$, the midpoints of $AB, BC, CD, DA$ are $L, M, P, Q$. Prove that $LMPQ$ is a parallelogram.

In the quadrilateral $ABCD$ the midpoints of $AB, BC, CD, DA$ are $L, M, P, Q$. If it is also given that the diagonals $AC$ and $BD$ are equal. What further statement can be made about the parallelogram $LMPQ$?

$ABCD$ is a parallelogram, $E$ is the midpoint of $AB$ and $F$ is the midpoint of $CD$. $GH$ is any line that intersects $AD, EF$ and $BC$ in $G, P$ and $H$ respectively. Prove that $GP=PH$.

Prove that in a parallelogram, the lines joining a pair of opposite vertices to the mid-points of a pair of opposite sides trisect a diagonal.

In the figure given below, $E$ is the midpoint of side $AD$ of a trapezium $ABCD$ with $AB\Vert DC.$ A line through $E$ parallel to $AB$, meets $BC$ in $F$. Show that $F$ is the midpoint of $BC$.