Advertisements
Advertisements
Question
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
Show that AE and DF bisect each other.
Advertisements
Solution
Since D and F are mid-points of AB and AC, by Mid-point theorem,
BC = 2DF
Now,
BC = BE + EC
DF = DG + GF
But E is the mid-point of BC,
⇒ BE = EC ....(i)
Also, AG = GE ....(G is the mid-point of AE)
Consider ΔABE and ΔACE, by mid-point theorem,
BE = 2DG and EC = 2GF
⇒ 2DG = 2GF ....[From (i)]
⇒ DG = GF
Hence, AE and DF bisect each other.
APPEARS IN
RELATED QUESTIONS
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral
bisect each other.
In the given figure, points X, Y, Z are the midpoints of side AB, side BC and side AC of ΔABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
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; 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 ∠FDB if ∠ACB = 115°.
In parallelogram PQRS, L is mid-point of side SR and SN is drawn parallel to LQ which meets RQ produced at N and cuts side PQ at M. Prove that M is the mid-point of PQ.
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.
Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square is also a square.
AD is a median of side BC of ABC. E is the midpoint of AD. BE is joined and produced to meet AC at F. Prove that AF: AC = 1 : 3.
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.
