In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

Question

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that : (i) AD is perpendicular to FE. (ii) AD and FE bisect each other.

KnowledgeBoat · Accepted Answer

(i) Given: In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB. To prove: AD is perpendicular to FE. Construction: Draw line AD which meets BC at D and join F and E. M is the intersection point of AD and FE. Join ED and FD. Proof: In △ABC, AB = AC ⇒ AB = AC ⇒ AF = AE ..................(1) E is mid point of AC and F is mid point of AB. By mid-point theorem, EF ∥ BC and EF = BC or, EF ∥ BD and EF = BD Similarly, E is mid point of AC and D is mid point of BC. By mid-point theorem, ED ∥ AB and ED = AB or, ED ∥ AF and ED = AF D is mid point of BC and F is mid point of AB. By mid-point theorem, DF ∥ AC and DF = AC or, DF ∥ AE and DF = AE From the midpoint theorem, the quadrilateral AEDF has opposite sides equal and parallel, thus it forms a parallelogram. Since AE = AF, this parallelogram is a rhombus because all sides are equal. In a rhombus, the diagonals bisect each other at right angles (90°). Therefore, AD and EF bisect each other at 90°. Hence, AD is perpendicular to FE. (ii) Since AEDF is a rhombus (as shown in part (i)), we know that the diagonals of a rhombus bisect each other. Thus, AD and FE bisect each other at their point of intersection(M). Hence, AD and FE bisect each other.

Mathematics

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

Mid-point Theorem

Related Questions

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

In the following figure, l // m // n. If OC = OD = 5 cm, OA = 8cm and OE = 10 cm, find OB and OF.

Mathematics

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :(i) AD is perpendicular to FE.(ii) AD and FE bisect each other.

Mid-point Theorem

Related Questions

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

In the following figure, l // m // n. If OC = OD = 5 cm, OA = 8cm and OE = 10 cm, find OB and OF.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.