In $\triangle ABC; M$ is mid$-$point of $\text{AB}, N$ is mid$-$point of $\text{AC}$ and $D$ is any point in base $\text{BC}.$ Use the intercept Theorem to show that $\text{MN}$ bisects $\text{AD}.$
View full solution →Question types
Mid-point and Its Converse [Including Intercept Theorem] question types
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Sample Questions
Mid-point and Its Converse [Including Intercept Theorem] questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
In $\triangle ABC, D$ and $E$ are points on side $\text{AB}$ such that $\text{AD} = \text{DE} = \text{EB}.$ Through $D$ and $E,$ lines are drawn parallel to $\text{BC}$ which meet side $\text{AC}$ at points $F$ and $G$ respectively. Through $F$ and $G,$ lines are drawn parallel to $\text{AB}$ which meets side $\text{BC}$ at points $M$ and $N$ respectively. Prove that: $\text{BM} =\text{ MN} = \text{NC}.$
View full solution →In $\triangle ABC,$ angle $B$ is obtuse. $D$ and $E$ are mid$-$points of sides $\text{AB}$ and $\text{BC}$ respectively and $F$ is a point on side $\text{AC}$ such that $\text{EF}$ is parallel to $\text{AB}.$ Show that $\text{BEFD}$ is a parallelogram.
View full solution →In the figure, give below, $2\text{AD}=\text{AB}, P$ is mid$-$point of $\text{AB}, Q$ is mid$-$point of $\text{DR}$ and $\text{PR} \| \text{BS}.$ Prove that:$(i) \text{AQ} \| \text{BS},(ii) \text{DS} =3 Rs.$
View full solution →In $\triangle ABC, AD$ is the median and $DE$ is parallel to $BA,$ where $E$ is a point in $AC.$ Prove that $BE$ is also a median.
View full solution →In parallelogram $\text{ABCD, E}$ and $F$ are mid$-$points of the sides $AB$ and $CD$ respectively. The line segments $AF$ and $BF$ meet the line segments $ED$ and $EC$ at points $G$ and $H$ respectively.Prove that$:(i)$ Triangles $HEB$ and $FHC$ are congruent$;(ii)\text{GEHF}$ is a parallelogram.
View full solution →Use the following figure to find$:(i) BC$, if $AB =7.2 \ cm.(ii) G E$, if $F E=4 \ cm.(iii) A E$, if $B D=4.1 \ cm(iv) D F$, if $C G=11 \ cm$.
View full solution →In $\triangle ABC ; D$ and $E$ are mid$-$points of the sides $AB$ and $AC$ respectively. Through $E$, a straight line is drawn parallel to $AB$ to meet $BC$ at $F.$Prove that $\text{BDEF}$ is a parallelogram. If $AB = 16 \ cm, AC = 12 \ cm$ and $BC = 18 \ cm$,find the perimeter of the parallelogram $\text{BDEF}.$
View full solution →Prove that the figure obtained by joining the mid$-$points of the adjacent sides of a rectangle is a rhombus.
View full solution →$D$ and $F$ are mid$-$points of sides $AB$ and $AC$ of a $ \triangle ABC.$ A line through $F$ and parallel to $AB$ meets $BC$ at point $E$.
View full solution →- Prove that BDFE is a parallelogram
- Find AB, if EF = 4.8 \ cm.
If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral $\text{ABCD}$ is a rectangle,show that the diagonals $AC$ and $BD$ intersect at the right angle.
View full solution →In $\triangle ABC$, the medians $BP$ and $CQ$ are produced up to points $M$ and $N$ respectively such that $BP = PM$ and $CQ = QN$. Prove that:
View full solution →- $M, A$, and $N$ are collinear.
- $A$ is the mid$-$point of $MN.$
The side $AC$ of a $\triangle ABC$ is produced to point $E$ so that $CE =AC. D$ is the mid$-$point of $BC$ and $ED$ produced meets $AB$ at $F$. Lines through $D$ and $C$ are drawn parallel to $AB$ which meet $AC$ at point $P$ and $EF$ at point $R$ respectively. Prove that: $(i) 3DF = EF;(ii) 4CR = AB.$
View full solution →Adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of the sides of this parallelogram. Show that its diagonals are in ratio $\sqrt{3}:1.$
View full solution →A parallelogram $\text{ABCD}$ has $P$ the mid$-$point of $D C$ and $Q$ a point of Ac such that
$CQ =\frac{1}{4} AC . PQ$ produced meets $BC$ at $R$.
Prove that
$(i) R$ is the midpoint of $B C$
$(ii) PR =\frac{1}{2} DB$
View full solution →$CQ =\frac{1}{4} AC . PQ$ produced meets $BC$ at $R$.
Prove that
$(i) R$ is the midpoint of $B C$
$(ii) PR =\frac{1}{2} DB$
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