Question

area of parallelogram abcd is x cm square .if EFG and H are mid-points of the sides,find the area of EFGH

Answer 1

Given : A parallelogram ABCD · E, F, G, H are mid-points of sides AB, BC, CD, DA respectively

To Prove : ar (EFGH) = 1/ 2 ar (ABCD)

Construction : Join AC and HF. Proof : In ∆ABC,

E is the mid-point of AB. F is the mid-point of BC. ⇒ EF is parallel to AC and EF = 1 2 AC ... (i)

Similarly, in ∆ADC, we can show that HG || AC and HG = 1 2 AC ...

(ii) From (i) and (ii) EF || HG and EF = HG ∴ EFGH is a parallelogram.

[One pour of opposite sides is equal and parallel]

In quadrilateral ABFH, we have HA = FB and HA || FB

[AD = BC ⇒ 1/ 2 AD = 1/ 2 BC ⇒ HA = FB] ∴

ABFH is a parallelogram.

[One pair of opposite sides is equal and parallel]

Now, triangle HEF and parallelogram HABF are on the same base HF and between the same parallels HF and AB.

∴ Area of ∆HEF = 1/ 2 area of HABF ...

(iii) Similarly, area of ∆HGF = 1/ 2 area of HFCD .

.. (iv) Adding (iii) and (iv), Area of ∆HEF + area of ∆HGF = 1/ 2

(area of HABF + area of HFCD)

$\implies$ ar (EFGH) = 1/ 2 ar (ABCD)

$\bf\red{ \:Hence \: it's \: Proved}$

Answer 2

Answer:

https://www.toppr.com/ask/question/if-e-f-g-and-h-are-respectively-the-midpoints/#acceptedAnswer

Step-by-step explanation:

This one is aafe