In △ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.
Mid-point Theorem
2 Likes

Question

In △ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.

Mid-point Theorem

2 Likes

KnowledgeBoat · Accepted Answer

Since, D and E are mid-points of AB and AC respectively, DE || BC or DE || BF and DE = BC ......(By midpoint theorem) .....(i) Given, through E, a straight line is drawn parallel to AB to meet BC at F. F will be mid-point of BC (By converse of mid-point theorem). Since, F is midpoint of BC, ∴ BF = BC .....(ii) From (i) and (ii) we get, DE = BF and DE || BF. Since, F and E are mid-points of BC and AC respectively, FE || AB or FE || BD and FE = AB ......(By midpoint theorem) .....(iii) Since, D is midpoint of AB, ∴ BD = AB .....(iv) From (iii) and (iv) we get, BD = FE and BD || FE. Since, DE = BF, DE || BF and BD = FE, BD || FE Hence, proved that BDEF is a parallelogram. Perimeter of BDEF = BD + DE + FE + BF = BD + DE + BD + FE = 2(BD + FE). BD = AB = = 4 cm. FE = BC = = 4.5 cm. Perimeter of BDEF = 2(BD + FE) = 2(4 + 4.5) = 2 × 8.5 = 17 cm. Hence, perimeter of BDEF = 17 cm.

Mathematics

In △ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.

Mid-point Theorem

Related Questions

If D, E and F are mid-points of the sides BC, CA and AB respectively of a △ABC, prove that AD and FE bisect each other.

The diagonals of a quadrilateral ABCD are perpendicular. Show that the quadrilateral formed by joining the mid-points of its adjacent sides is a rectangle.

In the adjoining figure, ABCD is a parallelogram and E is mid-point of AD. DL || EB meets AB produced at F. Prove that B is mid-point of AF and EB = LF.