Mid Point Theorem

Find the length of the median of the given triangle $PQR$ whole sides are given as follows, $PQ=10 \mathrm{units}, PR=13 \mathrm{units}$ and $QR=8 \mathrm{units}$, respectively, in which $PM$ is the median formed on side $QR$.

If the length of each median of an equilateral triangle is $4 \mathrm{cm}$ then the perimeter of the triangle is

Show that the median of a triangle divides it into two triangles of equal areas

Prove that in an isosceles triangle the perpendicular drawn from the vertex angle to the base bisect the vertex angle and the base. 

In figure, $\mathrm{G}$ is the point of concurrence of medians of $∆\mathrm{DEF}$. Take point $\mathrm{H}$ on ray such that $\mathrm{D}-\mathrm{G}-\mathrm{H}$ and $\mathrm{DG}=\mathrm{GH}$, then prove that $\mathrm{GEHF}$ is a parallelogram.

In $∆\mathrm{ABC}, \mathrm{E}$ is the mid-point of median $\mathrm{AD}$ such that $\mathrm{BE}$ produced meets $\mathrm{AC}$ at $\mathrm{F}$. If $\mathrm{AC}=10.5 \mathrm{cm}$, then $\mathrm{AF}=3.5 \mathrm{cm}$.

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

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

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

In a parallelogram $ABCD,E,$and $F$ are the midpoints of the sides $AB$ and $DC$ respectively. Show that the line segment $AF$ and $EC$ trisect the diagonal $BD$.

Show that the figure formed by joining the midpoints of sides of a rhombus successively is a rectangle.

$ABCD$ is quadrilateral $E, F, G$ and $H$ are the midpoints of $AB, BC, CD$ and $DA$ respectively. Prove that $EFGH$ is a parallelogram.

$ABC$ is a triangle. $D$ is a point on $AB$ such that $AD=\frac{1}{4}AB$ and $E$ is the point on $AC$ such that $AE=\frac{1}{4}AC$. If $DE=2 cm$ find $BC$.

If $D$ and $E$ are respectively the midpoints of the sides $AB$ and $BC$ of $∆ABC$ in which $AB=7.2 \mathrm{cm}, BC=9.8 \mathrm{cm}$ and $AC=3.6 \mathrm{cm}$ then determine the length of $DE$.

In the design below, assume there are 5 rectangles and 5 rhombuses. The 1st quadrilateral is a rectangle.

Which of the following numbers represents the innermost rectangle?

Assume there are 10 quadrilaterals in the design below.

Let $ {\text{A}}_{1}$ and $ {\text{A}}_{9}$ be the areas of the 1st and the 9th quadrilaterals.

Which of these formulae gives the correct relationship between $ {\text{A}}_{1}$ and $ {\text{A}}_{9}$?

In the figure below, area of the rectangle is ‘A’. The midpoints of its sides are joined to get a rhombus.

What is the area of the rhombus inside the rectangle?