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Grade: 10

                        

if the diagonals of a quadrilateral abcd meet o, then prove that abc: adc=bo:od

if the diagonals of a quadrilateral abcd meet o, then prove that abc: adc=bo:od

11 months ago

Answers : (2)

Arun
25768 Points
							
Given
 
ABCD is a parallelogram
 
AC and BD are diagonals
 
O is a mid point
 
To prove BO =OD
 
O=o common point
 
Angle b=angle d opp angles in ll gm are                     equal
 
Ob=od sides opp to equal angles are equal
 
Hence proved
11 months ago
Aditya Gupta
2075 Points
							
note that aruns ans above is absurd and wrong coz he has assumed abcd to be a || gm, even though it is nowhere mentioned in the ques. we will prove the result correctly for any quad abcd as follows:
first note that area of any triangle PQR= ½ PQ*PR*sin(angle QPR) (this is a std formula nd u can find multiple proofs of this online too!)
also, let angle AOB= x, whence angle DOC= x and angle AOD= angle BOC= pi – x.
further AO= a, BO= b, CO= c and DO= d
now, area(ABC)= area(AOB) + area(BOC)= ½ ab*sinx + ½ bc*sin(pi – x)= ½ b(a+c)*sinx since sinx= sin(pi – x)
similarly area(ADC)= area(AOD) + area(COD)= ½ ad*sin(pi – x) + ½ cd*sinx= ½ d(a+c)*sinx
hence, area(ABC)/area(ADC)= ½ b(a+c)*sinx / ½ d(a+c)*sinx 
= b/d
= BO/DO
KINDLY APPROVE :))
11 months ago
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