State true or false.Diagonals AC and BD of a quadrilateral

ABCD
intersect each other at

P
intersect each other at . Then

ar

ΔAPB
×ar

ΔCPD
=ar

ΔAPD
×ar

ΔBPC
.
. Then

Question

State true or false.Diagonals AC and BD of a quadrilateral

ABCD
  intersect each other at

P
  intersect each other at . Then

ar
    
     ΔAPB
   ×ar
    
     ΔCPD
   =ar
    
     ΔAPD
   ×ar
    
     ΔBPC
   .
 . Then

Accepted Answer

Given that, diagonals AC and BD of a quadrilateral

ABCD
  intersect each other at

P
 .Draw a quadrilateral

ABCD
 .Draw

AM⊥BD
  And

CN⊥BD
 . Therefore, the figure is as follows,Compute that area of

ΔAPB
  and

ΔAPD.
  Area of a triangle

=
    1
    2
   
   × base × height.
  The area of

ΔAPB
  can be found as,Area of

ΔAPB=
    1
    2
   
   ×BP×AM.....
    1
   
  Also, the area of

ΔAPD
  can be found as,Area of

ΔAPD=
    1
    2
   
   ×DP×AM.....(2)
 Compute that area of

ΔCPD
 and

ΔBPC
 .Area of a triangle

=
    1
    2
   
   × base × height.
  The area of

ΔCPD
  can be found as,Area of

ΔCPD=
    1
    2
   
   ×DP×CN......
    3
   
  Also, the area of  can be found as,Area of

ΔBPC=
    1
    2
   
   ×BP×CN......(4)
 Compute that

ar
    
     △APB
   ×ar
    
     △CPD
   =ar
    
     △APD
   ×ar
    
     △BPC
   .
  That is,Taking left hand side from equation (

1
 ) and (

3
 ), it can be written as

ar
    
     ΔAPB
   ×ar
    
     ΔCPD
   =(
    1
    2
   
   ×BP×AM)×(
    1
    2
   
   ×DP×CN)

=
    1
    2
   
   ×BP×AM×
    1
    2
   
   ×DP×CN
  . . . . . . (5) Taking RHS from equation (

2
 ) and (

4
 ), it can be written as,

ar
    
     ΔAPD
   ×ar
    
     ΔBPC
   =(
    1
    2
   
   ×DP×AM)×(
    1
    2
   
   ×BP×CN)

=
    1
    2
   
   ×DP×AM×
    1
    2
   
   ×BP×CN
  . . . . . . (6) Therefore, from (5) and (6),LHS = RHSThat is,

ar
    
     ΔAPB
   ×ar
    
     ΔCPD
   =ar
    
     ΔAPD
   ×ar
    
     ΔBPC
   
 .Hence, the statement is true.Therefore, option 1 is correct.

State true or false.

Diagonals AC and BD of a quadrilateral $A B C D$ intersect each other at $P$ intersect each other at . Then $a r Δ A P B × a r Δ C P D = a r Δ A P D × a r Δ B P C .$ . Then

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State true or false.Diagonals AC and BD of a quadrilateral ABCD intersect each other at P intersect each other at . Then ar ΔAPB ×ar ΔCPD =ar ΔAPD ×ar ΔBPC . . Then