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: Q A and P are collinear.
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)
In ΔABC,
∠ABC + ∠ACB ++ ∠BAC = 180°
⇒ ∠DBC + ∠ECB + ∠BAC = 180°
⇒ ∠DAQ + ∠EAP + ∠BAC = 180° ...[From (i) and (iii)]
⇒ Q, A, P are collinear.
APPEARS IN
RELATED QUESTIONS
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.
Let Abc Be an Isosceles Triangle in Which Ab = Ac. If D, E, F Be the Mid-points of the Sides Bc, Ca and a B Respectively, Show that the Segment Ad and Ef Bisect Each Other at Right Angles.
Fill in the blank to make the following statement correct
The triangle formed by joining the mid-points of the sides of an isosceles triangle is
L and M are the mid-point of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.
Use the following figure to find:
(i) BC, if AB = 7.2 cm.
(ii) GE, if FE = 4 cm.
(iii) AE, if BD = 4.1 cm
(iv) DF, if CG = 11 cm.
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, 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, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
RT = `(1)/(3)"PQ"`
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if ______.
D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
