Advertisements
Advertisements
Question
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.
Advertisements
Solution
In ΔBDC and ΔADQ,
CD = DQ ....(given)
∠BDC = ∠ADQ ....(vertically opposite angles)
BD = AD ....(D is the mid-point of AB)
∴ ΔBDC ≅ ΔADQ
⇒ ∠DBC = ∠DAQ (c.p.c.t)....(i)
And, BC = AQ (c.p.c.t)....(ii)
Similarly, we can prove ΔCEB ≅ ΔAEP
⇒ ∠ECB = ∠EAP (c.p.c.t)....(iii)
And, BC = AP (c.p.c.t)....(iv)
From (ii) and (iv),
AQ = AP
⇒ A is the mid-point of PQ.
APPEARS IN
RELATED QUESTIONS
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 Fig. below, triangle ABC is right-angled at B. Given that AB = 9 cm, AC = 15 cm and D,
E are the mid-points of the sides AB and AC respectively, calculate
(i) The length of BC (ii) The area of ΔADE.
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 the Figure, `square`ABCD is a trapezium. AB || DC. Points P and Q are midpoints of seg AD and seg BC respectively. Then prove that, PQ || AB and PQ = `1/2 ("AB" + "DC")`.
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: QAP is a straight line.
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
ABCD is a parallelogram.E is the mid-point of CD and P is a point on AC such that PC = `(1)/(4)"AC"`. EP produced meets BC at F. Prove that: F is the mid-point of BC.
The diagonals AC and BD of a quadrilateral ABCD intersect at right angles. Prove that the quadrilateral formed by joining the midpoints of quadrilateral ABCD is a rectangle.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
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.
