In the adjoining figure, ABCD is a parallelogram in which E is the mid-point of DC and F is a point on AC such that CF = AC. If EF is produced to meet BC in G, prove that G is the mid-point of BC.

Question

KnowledgeBoat · Accepted Answer

Join BD. Let BD intersect AC at point O. We know that, The diagonals of a parallelogram bisect each other. AO = CO and OD = OB Given, ⇒ CF = AC ⇒ CF = (AO + CO) ⇒ CF = (CO + CO) ⇒ CF = 2 CO ⇒ CF = CO ∴ F is the mid-point of CO. 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 equal to half of it. In △COD, Since, E and F are the mid-points of DC and OC respectively. EF || OD Since, EG and BD are straight lines. Thus, FG || OB By converse of mid-point theorem, The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side. In △COB, Since, F is the mid-point of CO and FG || OB ∴ G is the mid-point of BC. Hence, proved that G is the mid-point of BC.

Mathematics

In the adjoining figure, ABCD is a parallelogram in which E is the mid-point of DC and F is a point on AC such that CF = AC. If EF is produced to meet BC in G, prove that G is the mid-point of BC.

Mid-point Theorem

Related Questions

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

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.

Mathematics

In the adjoining figure, ABCD is a parallelogram in which E is the mid-point of DC and F is a point on AC such that CF = AC. If EF is produced to meet BC in G, prove that G is the mid-point of BC.

Mid-point Theorem

Related Questions

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

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.

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.