Advertisements
Advertisements
Question
Side AC of a ABC is produced to point E so that CE = `(1)/(2)"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 meets AC at point P and EF at point R respectively. Prove that: 3DF = EF
Solution
In ΔBDF and ΔDRC,
BD = DC ...(D is the mid-point of BC)
CR || PD || AB
∠BFD = DRC ...(alternate angles)
∠BDF = RDC ...(vertivally opposite angles)
Therefore,
ΔBDF ≅ ΔDRC
⇒ DF = DR .....(i)
In ΔABC,
D is the mid-point of BC and DP || AB
Therefore, P is the mid-point of AC.
In ΔDEP,
C is the mid-point of PE and DP || RC || AB ...(CE = `(1)/(2)"AC"` and P is the mid-point of AC)
Therefore, R is the mid-point of DE.
⇒ DR = RE ......(ii)
But EF = DF + DR + RE
EF = DF + DF + DF
EF = 3DF.
APPEARS IN
RELATED QUESTIONS
ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH
Fill in the blank to make the following statement correct:
The triangle formed by joining the mid-points of the sides of a right triangle is
ABCD is a quadrilateral in which AD = BC. E, F, G and H are the mid-points of AB, BD, CD and Ac respectively. Prove that EFGH is a rhombus.
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:
- M, A, and N are collinear.
- A is the mid-point of MN.
In the given figure, AD and CE are medians and DF // CE.
Prove that: FB = `1/4` AB.
In ΔABC, D is the mid-point of AB and E is the mid-point of BC.
Calculate:
(i) DE, if AC = 8.6 cm
(ii) ∠DEB, if ∠ACB = 72°
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
If AE and DF intersect at G, and M and N are the midpoints of GB and GC respectively, prove that DMNF is a parallelogram.
The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if, ______.
E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC. [Hint: Join AC]
D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆DEF is also an equilateral triangle.
