State true or false.In a

ΔABC
,

P
, and

Q
and are respectively the mid-point of

AB
are respectively the mid-point of and

BC
and .

R
. is the mid-point of

AP
is the mid-point of . Then

ar(ΔPBQ)=ar(ΔARC)
. Then .

Question

State true or false.In a

ΔABC
 ,

P
 ,  and

Q
  and  are respectively the mid-point of

AB
  are respectively the mid-point of  and

BC
  and .

R
 .  is the mid-point of

AP
  is the mid-point of . Then

ar(ΔPBQ)=ar(ΔARC)
 . Then .

Accepted Answer

Given that,

P
  and

Q
  are respectively the midpoint of

AB
  and

BC
 . And

R
  is the midpoint of

AP
 .We know that, median divides the triangle into two triangles of equal area.PC is the median of

ΔABC
 , so it divides into 2 triangles of equal areas.

∴ar
    
     ΔBPC
   =ar
    
     ΔAPC
   
                                 ......(1)RC is the median of

ΔAPC
 .

∴ar
    
     ΔARC
   =
    1
    2
   
   ar
    
     ΔAPC
   
                               ......(2)From (1) and (2),

ar
    
     ΔBPC
   =2ar
    
     ΔARC
   
 .    ……(3)PQ is the median of

ΔBPC
 .

∴ar
    
     ΔPBQ
   =
    1
    2
   
    ar
    
     ΔBPC

∴2ar
    
     ΔPBQ
   =ar
    
     ΔBPC
   
                                  ......(4)From equation (3) and (4), we get,

ar
    
     ΔPBQ
   =ar
    
     ΔARC
   
 .                                   ......(5)Hence, the statement is true.Hence, option 1) is correct.

State true or false.

In a $Δ A B C$ , $P$ , and $Q$ and are respectively the mid-point of $A B$ are respectively the mid-point of and $B C$ and . $R$ . is the mid-point of $A P$ is the mid-point of . Then $ar (Δ PBQ) = ar (Δ ARC)$ . Then .