Question

ABCD is a quadrilateral. The diagonals of ABCD intersect at the point P.The area of the triangles APD and BPC are 27 and 12 respectively.If the areas of triangles APB and CPD are equal then find the area of triangle APB.

Answer 1

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>>Diagonals AC and BD of a quadrilateral A

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Diagonals AC and BD of a quadrilateral ABCD intersect each other at P show that

ar.(△APB)×ar.(△CPD)=ar.(△APD)×ar.(△BPC)

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Solution

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Data: Diagonals AC and BD of a quadrilateral ABCD intersect each other at P.

To Prove: ar.(△APB)×ar.(△CPD)=ar.(△APD)×ar.(△BPC)

Construction: Draw AM⊥DB,CN⊥DB

Proof: ar.(△APB)×ar.(△CPD)=

=(

2

1

×PB×AM)×(

2

1

×PD×CN)

=

4

1

×PB×AM×PD×CN....(i)

ar.(△APD)×ar.(△BPC)=

=(

2

1

×PD×AM)×(

2

1

×PB×CN)

=

4

1

×PD×AM×PB×CN....(ii)

From (i) and (ii)

ar.(△APB)×ar.(△CPD)=ar.(△APD)×ar.(△BPC)