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ABCD is a quadrilateral and P,Q are midpoints of CD,AB,AP,DQ meet at X, and BP,CQ meet at Y.prove that (area of triangle ADX+area of triangle BCY )=area of quadrilateral PXQY.

ABCD is a quadrilateral and P,Q are midpoints of CD,AB,AP,DQ meet at X, and BP,CQ meet at Y.prove that (area of triangle ADX+area of triangle BCY )=area of quadrilateral PXQY.

Grade:Upto college level

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

 

Hi Phani,

 

This is a good problem, which will give you how to approach problems where multiple concepts have to be used.

 

 

Hint: To prove the similar triangles result, extend CD to meat AB in A'

 

A'D/A'P = h1/h2

====>  A'D/(A'D+DP) = h1/h2

ie DP/A'D = h2/h1 - 1

 

and A'D/A'C = h1/h3

====>  A'D/(A'D+2DP) = h1/h3

ie 2DP/A'D = h3/h1 - 1

 

hence 2*(h2/h1 - 1) = h3/h1 - 1

2*(h2 - h1) = h3 - h1

therefore, 2*h2 = h1 + h3.

 

That solves the problem.

All the Best.

 

Regards,

Ashwin (IIT Madras)

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