Advertisements
Advertisements
प्रश्न
If L and M are the mid-points of AB, and DC respectively of parallelogram ABCD. Prove that segment DL and BM trisect diagonal AC.
Advertisements
उत्तर
Since L and M are the mid-points of AB and Dc respectively.
`"BL" = (1)/(2)"AB" and "Dm" = (1)/(2)"DC"`....(i)
But ABCD is a parallelogram
Therefore, AB = CD and AB || DC
⇒ BL = DM and BL || Dm ...(from (i))
⇒ BLDM is a parallelogram.
⇒ DL || Dm
⇒ LP || BQ ............(ii)
It is known that the segment drawn through the mid-point of one side of a triangle and parallel to the other side bisects the third side.
In ΔABQ , L is the mid-point of AB and MQ || PD
Therefore, P is mid-point of AQ
Hence, AP = PQ ..........(iii)
Similarly, in ΔCPD, M is the mid-point of CD and LP || BQ
Therefore, Q is mid-point of CP
Hence, PQ = QC ..........(iv)
From (iii) and (iv)
AP = PQ = QC
Therefore, P and Q trisect AC
Thus, DL and BM trisect AC.
APPEARS IN
संबंधित प्रश्न
ABCD is a square E, F, G and H are points on AB, BC, CD and DA respectively, such that AE = BF = CG = DH. Prove that EFGH is a square.
In a ∆ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are 7 cm, 8 cm and 9 cm, respectively, find the perimeter of ∆DEF.
In the given figure, `square`PQRS and `square`MNRL are rectangles. If point M is the midpoint of side PR then prove that,
- SL = LR
- LN = `1/2`SQ
In triangle ABC, P is the mid-point of side BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R.
Prove that : (i) AP = 2AR
(ii) BC = 4QR
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:
- 3DF = EF
- 4CR = AB
In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and G, lines are drawn parallel to AB which meets side BC at points M and N respectively. Prove that: BM = MN = NC.
In ΔABC, P is the mid-point of BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R. Prove that: BC = 4QR
In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: A is the mid-point of PQ.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. Show that AF = `1/3` AC.
