CBSE Class 9 Answered

ABCD is a trapezium with parallel sides AB=a cm and DC=b cm .E and F are midpoint of the non parallel sides.find the ratio of area [ABFE] and area [EFCD].

SOLVE IT.

ALSO,explain how EF= a+b/2.

Asked by ranjitrk1 | 14 Mar, 2016, 10:16: PM

Expert Answer

$Extend space AB space to space meet space CE space at space straight G. In space increment EAG space and space increment ECD comma angle AEG approximately equal to angle CED space left parenthesis Vertically space opposite space angles right parenthesis angle CDE approximately equal to angle GAE space left parenthesis since space CD parallel to AB space that space is space CD parallel to GA comma space so space angles space are space alternate right parenthesis ED space approximately equal to EA space left parenthesis Since space straight E space is space the space mid minus point space of space AD. right parenthesis rightwards double arrow increment EAG space approximately equal to space increment ECD space space left parenthesis By space ASA space congruence space criterion right parenthesis So comma space EG approximately equal to EC comma space that space is space straight E space is space the space mid minus point space of space GC and space AG approximately equal to CD rightwards double arrow AG space equals space CD space equals space straight a In space increment CGB comma space straight E space and space straight F space are space the space mid minus points space of space CG space and space CB space respectively comma So comma space EF space parallel to space GB comma space that space is space EF space parallel to space AB space and space EF space equals space 1 half GB SImilarly space we space can space prove space EF space parallel to CD. Now comma space EF space equals space 1 half GB rightwards double arrow space EF space equals space 1 half left parenthesis AG space plus space AB right parenthesis rightwards double arrow space EF space equals space 1 half left parenthesis straight a space plus straight b right parenthesis space space space. space. space. space. space. left parenthesis Since space AG space equals space CD space equals space straight a right parenthesis Since space straight E space is space the space mid minus point space of space AD space and space EF space is space parallel space to space AB comma space if space 2 straight h space is space the space height space between space CD space and space AB comma space then space straight h space is space the space height space between space CD space and space EF space and space straight h space is space the space height space between space EF space and space AB. space Now comma space fraction numerator straight A left parenthesis ABEF right parenthesis over denominator straight A left parenthesis EFCD right parenthesis end fraction equals space fraction numerator begin display style 1 half left parenthesis EF space plus AB right parenthesis straight h end style over denominator 1 half left parenthesis EF space plus CD right parenthesis straight h end fraction equals fraction numerator begin display style 1 half left parenthesis straight a space plus straight b right parenthesis space plus straight b end style over denominator 1 half left parenthesis straight a space plus straight b right parenthesis space plus straight a end fraction equals fraction numerator straight a space plus space straight b space plus space 2 straight b over denominator straight a space plus space straight b space plus space 2 straight a end fraction equals fraction numerator straight a space plus space 3 straight b over denominator 3 straight a space plus space straight b end fraction Hence comma space fraction numerator straight A left parenthesis ABEF right parenthesis over denominator straight A left parenthesis EFCD right parenthesis end fraction equals fraction numerator straight a space plus space 3 straight b over denominator 3 straight a space plus space straight b end fraction space and space space EF space equals space 1 half left parenthesis straight a space plus straight b right parenthesis.$