Mid-Point and Intercept Theorems

In $∆ABC, P,Q$ and $R$ are the midpoint of $BC, CA$ and $AB$  respectively. If $BP=3.6 \mathrm{cm}$, $AC=4.8 \mathrm{cm}$ and $PQ=2.3 \mathrm{cm}$, find the value of $AB$.

In $∆\mathrm{ABC}, \mathrm{P},\mathrm{Q}$ and $\mathrm{R}$ are the midpoint of $\mathrm{BC}, \mathrm{CA}$ and $\mathrm{AB}$  respectively. If $\mathrm{BP}=3.6 \mathrm{cm}$, $\mathrm{AC}=4.8 \mathrm{cm}$ and $\mathrm{PQ}=2.3 \mathrm{cm}$, if the value of $\mathrm{AR}=k \mathrm{cm}$ then find $k$.

In $∆ABC, P,Q$ and $R$ are the midpoint of $BC, CA$ and $AB$  respectively. If $BP=3.6 \mathrm{cm}$, $AC=4.8 \mathrm{cm}$ and $PQ=2.3 \mathrm{cm}$, find the value of $RP$.

In $∆ABC, P,Q$ and $R$ are the midpoint of $BC, CA$ and $AB$  respectively. If $BP=3.6 \mathrm{cm}$, $AC=4.8 \mathrm{cm}$, $PQ=2.3 \mathrm{cm}$ and $RQ=x \mathrm{cm}$, then find the value of $x$.

In a $△ABC$, the medians $BP$ and $CQ$ are produced to points $M$ and $N$, respectively such that $BP=PM$ and $CQ=QN$. Prove that $A, M$ and $N$ are collinear.

$ABCD$ is a trapezium with $AB\parallel DC$ and $E$ is midpoint of $AD$. If $EF\parallel AB$ meets $BC at F$. Prove that $F$ is the midpoint of $BC$.

On $△\mathrm{ABC}$, $\angle \mathrm{B}$ is obtuse. $\mathrm{D} \& \mathrm{E}$ are the midpoints of $\mathrm{AB} \& \mathrm{BC}$ respectively and $\mathrm{F}$ is a point on $\mathrm{AC}$ such that $\mathrm{EF}\parallel \mathrm{AB}$. Prove that $\mathrm{BEFD}$ is a parallelogram.

In a $△ABC$, the medians $BP$ and $CQ$ are produced to points $M$ and $N$, respectively such that $BP=PM$ and $CQ=QN$. Prove that $A$ is the midpoint of $MN$.

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

The diagonals of a quadrilateral intersect at right angle. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is rectangle.

In parallelogram $PQRS$, $L$ is the mid point of side $SR$ and $SN$ is drawn parallel to $LQ$, which meets $RQ$ produced at $N$ and cuts side $PQ$ at $M$. Prove that $SN=2LQ$.

In parallelogram $PQRS$, $L$ is the mid point of side $SR$ and $SN$ is drawn parallel to $LQ$, which meets $RQ$ produced at $N$ and cuts side $PQ$ at $M$. Prove that $SP=\frac{1}{2}RN$.

In $∆ABC$, $P$ is the mid point of $BC, AR=RQ=QC$. Prove that $BR=4SR$.

In the given figure $ABCD$ is a trapezium in which $AB\parallel DC$. $P$ is the mid point of $AD$ and $PR\parallel AB$. Prove that: $PR=\frac{1}{2}(AB+CD)$.

$AD$ is a median of $∆ABC$, $E$ is the mid-point of $AD$. $BE$ produced meet $AC$ at $Q$. $DP\parallel BQ$. Prove that $BE:EQ=3:1$.

Prove that the four triangles formed by joining the mid-point of the sides are congruent to each other.

In the adjoining figure, $\angle A=\angle D=90°$, $\angle C=48°$ $BE$ bisector of $\angle B$. $AD$ and $BE$ intersect at $M$. Calculate $\angle AME$.

In the adjoining figure, $\angle A=\angle D=90°$, $\angle C=48°$. $BE$ bisector of $\angle B$, $AD$ and $BE$ intersect at $M$. Calculate $\angle BMD$.