Question

State true or false.

$L$ and $M$ are the mid-points of sides $AB$ and $DC$ respectively of parallelogram $ABCD$, then the segments $DL$ and $BM$ trisect diagonal $AC$.

• A. True
• B. False

Answer 1

The correct option is A True

Given: Parallelogram ABCD, L and M are midpoints of $AB$ and $DC$ respectively.

We know,
$AB=CD$
Hence, $DM=LB$ and $DM\parallel LB$ (As, $AB\parallel CD$)
Suppose $DL$ and $MB$ intersect AC at P and Q respectively.

Consider, $△ABC$,
$PL\parallel QB$
L is the midpoint of $AB$
Hence, P is midpoint of AQ (Converse of midpoint theorem)
Thus, $AP=PQ$ (1)

Similarly, in $△CDP$
$CQ=QP$ (2)

From (1) and (2)
$AP=PQ=CQ$
Hence, P and Q trisect diagonal $AC$.