In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :
(i) ∠EFG = 90°
(ii) If GH || FE, then H bisects CB.

Question

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that : (i) ∠EFG = 90° (ii) If GH || FE, then H bisects CB.

KnowledgeBoat · Accepted Answer

Join AC and BD, AC and BD intersects at O. (i) We know that, Diagonals of a kite intersect at right angles. ∠MON = 90° ...(1) By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ABD, E and F are mid-points of AB and AD, EF || BD and EF = BD By converse of mid-point theorem, The straight line drawn through the mid-point of one side side of a triangle parallel to another, bisects the third side. In △ABO, E is the mid-point of AB and EF || BD so, EM || BO, ∴ M is the mid-point of AO ⇒ AM = MO In △AOD, M and F are mid-points of AO and AD, MF || OD and MF = OD ...(1) In △ADC, G and F are mid-points of CD and AD, FG || AC and FG = AC (By mid-point theorem) In △AOD, F is the mid-point of AD and FG || AC so, FN || AO, ∴ N is the mid-point of OD (By converse of mid-point theorem) ⇒ ON = ND From eq.(1), we have: MF || OD and MF = OD ⇒ MF = ON ∴ OMFN is a parallelogram. We know that, Opposite angles of a parallelogram are equal. ⇒ ∠MON = ∠MFN = 90° From figure, ∠EFG = ∠MFN = 90° Hence, proved that ∠EFG = 90°. (ii) EF || BD (Proved above) A line through G is parallel to EF. ∴ GH || FE or GH || BD In △BCD, GH || BD and G is mid-point of CD. By converse of mid-point theorem, The straight line drawn through the mid-point of one side side of a triangle parallel to another, bisects the third side. ∴ H is the mid-point of BC (By converse of mid-point theorem) Hence, proved that if GH || FE, then H bisects CB.

Mathematics

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :
(i) ∠EFG = 90°
(ii) If GH || FE, then H bisects CB.

Mid-point Theorem

Related Questions

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

Mathematics

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :(i) ∠EFG = 90°(ii) If GH || FE, then H bisects CB.

Mid-point Theorem

Related Questions

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :
(i) ∠EFG = 90°
(ii) If GH || FE, then H bisects CB.