Q) Diagonals AC and BD of quadrilateral ABCD intersects at O . If AB=CD , then show that 1) ar(DOC) = ar(AOB) 2) ar (DCB) = ar(ACB) [ question from NCERT class 9 textbook.

In triangle ABD , AO is the median therefore ar(AOB) = ar(AOD) [ since median of a triangle divides it into 2 triangles with equal areas ]...................................(1) and also in triangle BCD , CO is the median and ar(COB) = ar ( COD) ...................................(2) . Adding (1) and (2) we get ar(ABD)= ar(BCD) . Thus diagonal divides it into 2 congruent triangles. Now we consider equal lines AB and CD after joining their end points we would either get an isosceles trapezium if the lines are not parallel or a parallelogram if the lines are parallel . But the diagonal of a isos. trapezium does not divide it into 2 triangles with equal areas , therefore it must be a parallelogram . Now triangles ACB and BCD are on the same base BC and between same parallels AD and BC . therefore ar(DCB)=ar(ACB) [ proof 2] and subtracting ar(BOC) from both sides we get ar(DOC)=ar(AOB] [ proof 1 ] . Hence proved . [ The question can easily be done by drawing perpendiculars from B and D on AC.]

## Comments

Sort by:

TopNewest[Adding (1) and (2) we get ar(ABD)= ar(BCD) ].................................... this part of your answer goes wrong because this part can be only be prooved if bo would be the median if BO would be the median than TRIANGLE BOC AND BOA would have equal area. and then ar[cob]=ar[boa]. Then area of all the four triangles would be equal. then you can prove your first part. so, this prove is wrong. – Abhishek Anand · 1 year, 10 months ago

Log in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

But , I have not used BO . I have just used median AO and CO .Log in to reply

– Abhishek Anand · 1 year, 10 months ago

but by using ao and co u can not prove ar[abd] =ar [dcb]. i think you are not clear with my answer let me explain . let us take ar[abd]=x and ar[cbd]=y then since ao is the median of x ar [aod]=ar[aob]=1/2x. since co is the median ar[cod]=ar[cob]=1/2y. and 1/2y would be only equal to 1/2x if their areas are equal that is do would be the median.i hope your doubt would be clear by this for more doubt about this contact me at 0788(2290223).Log in to reply

@karandeep singh ludhar @abhishek anand .. Kaddu's way is correct.. I did it the same way. This question appeared in our exams too right?.. Kaddu apna e-mail id de – Hrishik Mukherjee · 1 year, 10 months ago

Log in to reply

– Abhishek Anand · 1 year, 10 months ago

sir, tera answer sahi kiye kyaLog in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Hao.. Mere 2marks kate construction mei :'(Log in to reply

– Abhishek Anand · 1 year, 10 months ago

tu to median se kiya hai jo quetion me diya hua tha to tu construction kya kiyaLog in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

DB pe perpendiculars draw karke (From A and C) dono ko congruent proove krde.. Easy and simple =D.. Ye shayad question 17 thaa na?Log in to reply

– Abhishek Anand · 1 year, 10 months ago

vahi to mai bhi kiya hoo lekin kara ka prove galat haiLog in to reply

– Abhishek Anand · 1 year, 10 months ago

kaise sahi hai bhai /////explain??????????Log in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Search for NCERT Model Solutions :)Log in to reply

– Abhishek Anand · 1 year, 10 months ago

solution me perpendicular bana ke kiye haiLog in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

Kaise galat hai /////explain??????????Log in to reply

– Abhishek Anand · 1 year, 10 months ago

apna phone number de karta hoo explain\Log in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

But the question appeared in our exams just one time and also not right but in paper .Log in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Bhai english -_- .. Aur Kaddu Topper !!! IMO ka solve kr liya re :'( .. Rukk be dharta hoon terekoLog in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

Tereko ke baad bracket kon lagayega .Log in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Bracket konchika :p :p .. Tu aara hai FIITJEE?Log in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

Why is a tomato red and round . bataa.Log in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Bhai tu hi bta dey... Because Brinjal is long and purple :p _/_Log in to reply

– Abhishek Anand · 1 year, 10 months ago

hybrid brinjal khata hai kyaLog in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

Because if it would have been green and long it would be a cucumber . <english me isliye likha kyon ki question aur answer dono english me the .Log in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Stud ^^ _/_Log in to reply

– Abhishek Anand · 1 year, 10 months ago

tu apna number de naLog in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

Par maine kal hi to tujhe diya tha .Log in to reply

– Abhishek Anand · 1 year, 10 months ago

vapas se dede mere bhaiLog in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Kaddu ka 8959868806 hai.. I rememberLog in to reply

– Abhishek Anand · 1 year, 10 months ago

karan apne nani ke ghar gaya hai vaha ka number mang raha hooLog in to reply

– Karandeep Singh Ludhar · 1 year, 10 months ago

The number is not correct .HaaaaaaaaLog in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Bhai change ho gaya hoga.. Give me your new number and email idLog in to reply

– Hrishik Mukherjee · 1 year, 10 months ago

Ye question try karo.. Its a level 4 question by me : https://brilliant.org/problems/heart-attack-omg-p/?group=rb5lqcXVsIsB#!/solution-comments/89095/Log in to reply