Mathematics
In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.
Answer
Given,
D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Thus,
AD = DB, AF = FC and BE = EC
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.
Since, D and E are the mid-points of AB and BC respectively.
⇒ DE || AC
⇒ DE || AF …..(1)
Since, F and E are the mid-points of AC and BC respectively.
⇒ FE || AB
⇒ FE || AD …..(2)
In quadrilateral ADEF,
AD // FE and DE // AF
Since, opposite sides of quadrilateral ADEF are parallel.
∴ ADEF is a parallelogram.
Hence, proved that ADEF is a parallelogram.
Related Questions
In the given figure, D, E, F are the mid-points of the sides BC, CA and AB respectively.
(i) If AB = 6.2 cm, find DE
(ii) If DF = 3.8 cm, find AC
(iii) If perimeter of △ABC is 21 cm, find FE
In the given figure, LMN is a right triangle in which ∠M = 90°, P and Q are mid-points of LM and LN respectively. If LM = 9 cm, MN = 12 cm and LN = 15 cm, find :
(i) the perimeter of trapezium MNQP
(ii) the area of trapezium MNQP
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.
