Manipur Board Solutions for Chapter: Quadrilaterals, Exercise 2: EXERCISE 8.2

$ABCD$ is a rectangle and $P, Q, R, S$ are mid-points of the sides $AB, BC, CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rhombus.

$ABCD$ is a trapezium in which $AB\parallel DC$, $BD$ is a diagonal and $E$ is the midpoint of $AD$. A line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$. Show that $F$ is the midpoint of $BC$.

Show that the line segment joining the midpoints of the opposite sides of a quadrilateral bisect each other.

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of the hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that $D$ is midpoint of $AC$.

$ABC$ is a triangle right-angled at $C$. A line through the mid-point $M$ of the hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that $MD\perp AC$.

$ABC$ is a triangle right-angled at $C$. A line through the mid-point $M$ of the hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that $CM=MA=\frac{1}{2}AB$.

$ABC$ is an isosceles triangle in which $AB=AC$ and $D, E, F$ are midpoints of the sides $BC, CA, AB$ respectively. Prove that $AD$ and $EF$ bisect one another at right angles.

$ABCD$ is a trapezium in which $AB\parallel CD$. If $P$ and $Q$ are midpoints of $AD$ and $BC$ respectively, prove that $PQ\parallel AB$ and $PQ=\frac{1}{2}(AB+CD)$.