In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see Fig). Show that the line segments AF and EC trisect the diagonal BD.
ABCD is a parallelogram.
∠AB || CD
And hence, AE || FC
Again, AB = CD (Opposite sides of parallelogram ABCD)
AB= \(\frac{1}{2}\) CD= \(\frac{1}{2}\)
AE = FC (E and F are mid-points of side AB and CD)
In quadrilateral AECF, one pair of opposite sides (AE and CF) is parallel and equal to each other. Therefore, AECF is a parallelogram. ∠AF || EC (Opposite sides of a parallelogram)
In ∆DQC, F is the mid-point of side DC and FP || CQ (as AF || EC). Therefore, by using the converse of mid-point theorem, it can be said that P is the mid-point of
DQ.
∠DP = PQ ... (1)
Similarly, in ∆APB, E is the mid-point of side AB and EQ || AP (as AF || EC).
Therefore, by using the converse of mid-point theorem, it can be said that Q is the mid-point of PB.
∠PQ = QB ... (2)
From equations (1) and (2), DP =
PQ = BQ
Hence, the line segments AF and EC trisect the diagonal BD.
Use these adverbs to fill in the blanks in the sentences below.
awfully sorrowfully completely loftily carefully differently quickly nonchalantly
(i) The report must be read ________ so that performance can be improved.
(ii) At the interview, Sameer answered our questions _________, shrugging his shoulders.
(iii) We all behave _________ when we are tired or hungry.
(iv) The teacher shook her head ________ when Ravi lied to her.
(v) I ________ forgot about it.
(vi) When I complimented Revathi on her success, she just smiled ________ and turned away.
(vii) The President of the Company is ________ busy and will not be able to meet you.
(viii) I finished my work ________ so that I could go out to play