Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
In below Fig, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = `1/4` AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.
In triangle ABC; M is mid-point of AB, N is mid-point of AC and D is any point in base BC. Use the intercept Theorem to show that MN bisects AD.
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find FE, if BC = 14 cm
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find DE, if AB = 8 cm
In a parallelogram ABCD, E and F are the midpoints of the sides AB and CD respectively. The line segments AF and BF meet the line segments DE and CE at points G and H respectively Prove that: ΔGEA ≅ ΔGFD
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.
Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.
