Advertisements
Advertisements
Question
In ΔABC, BE and CF are medians. P is a point on BE produced such that BE = EP and Q is a point on CF produced such that CF = FQ. Prove that: A is the mid-point of PQ.
Advertisements
Solution
Since BE and CF are medians, F is the mid-point of AB and E is the mid-point of AC. Now, the line joining the mid-point of any two sides is parallel and half of the third side, we have In ΔACQ,
EF || AQ and EF = `(1)/(2)"AQ"` ....(i)
In ΔABP,
EF || AP and EF = `(1)/(2)"AP"` ....(ii)
From (i) and (ii)
`"EF" = (1)/(2)"AQ" and "EF" = (1)/(2)"AP"`
⇒ `(1)/(2)"AQ" = (1)/(2)"AP"`
⇒ AQ = AP
⇒ A is the mid-point of QP.
APPEARS IN
RELATED QUESTIONS
In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing
through A. If L is the mid-point of BC, prove that ML = NL.
ABC is a triang D is a point on AB such that AD = `1/4` AB and E is a point on AC such that AE = `1/4` AC. Prove that DE = `1/4` BC.
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.
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral
bisect each other.
In triangle ABC, AD is the median and DE, drawn parallel to side BA, meets AC at point E.
Show that BE is also a median.
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
In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P.
Prove that:
(i) BP = 2AD
(ii) O is the mid-point of AP.
ΔABC is an isosceles triangle with AB = AC. D, E and F are the mid-points of BC, AB and AC respectively. Prove that the line segment AD is perpendicular to EF and is bisected by it.
ABCD is a kite in which BC = CD, AB = AD. E, F and G are the mid-points of CD, BC and AB respectively. Prove that: ∠EFG = 90°
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.
