Question

The four triangle formed by joining the midpoints of the sides of a triangle respectively are

• A. similar, not necessarily congruent.
• B. congruent.
• C. equilateral.
• D. isosceles.

Answer 1

The correct option is B congruent.

$P,Q$ and $R$ are mid-points of sides $AB,BC$ and $AC$.

In $△PBQ$ and $△RQC$

$\Rightarrow$ $BQ=QC$ [ Mid-point theorem ]

$\Rightarrow$ $BP=RQ$ [ Mid-point theorem ]

$\Rightarrow$ $PQ=RC$ [ Mid-point theorem ]

$\therefore$ $△PBQ\cong △RQC$ [ By $SSS$ congruence theorem ] --- ( 1 )

In $△RQC$ and $△APR$

$\Rightarrow$ $RC=AR$ [ $R$ is the mid-point ]

$\Rightarrow$ $QC=PR$ [ Mid-point theorem ]

$\Rightarrow$ $AP=RQ$ [ Mid-point theorem ]

$\therefore$ $△RQC\cong △APR$ [ By $SSS$ congruence rule ] ---- ( 2 )

In $△APR$ and $△PQR$

$\Rightarrow$ $PR=PR$ [ Common side ]

$\Rightarrow$ $AP=RQ$ [ Mid-point theorem ]

$\Rightarrow$ $AR=PQ$ [ Mid-point theorem ]

$\therefore$ $△APR\cong △PQR$ [ By $SSS$ congruence rule ] ----- ( 3 )

From ( 1 ), ( 2 ) and ( 3 ) we get

$\Rightarrow$ $△PBQ\cong △RQC\cong △APR\cong PQR$

$\therefore$ The four triangle formed by joining the midpoints of the sides of a triangle respectively are congruent to each other.