State true or false.In the following figure,

ABC
is a right triangle right angled at

A
is a right triangle right angled at ,

BCED,ACFG
, and

ABMN
and are squares on the sides

BC,CA
are squares on the sides and

AB
and respectively. Line segment

AX⊥DE
respectively. Line segment meets

BC
meets at

Y
at . Then

ar(BYXD)=ar(ABMN)
. Then .

Question

State true or false.In the following figure,

ABC
  is a right triangle right angled at

A
  is a right triangle right angled at ,

BCED,ACFG
 ,  and

ABMN
  and  are squares on the sides

BC,CA
  are squares on the sides and

AB
 and  respectively. Line segment

AX⊥DE
  respectively. Line segment meets

BC
 meets  at

Y
  at . Then

ar(BYXD)=ar(ABMN)
 . Then .

Accepted Answer

Given that a figure.Consider,

ΔMBC,ΔABD
 .Since the sides

BC=BD
  of a square BCED are equal.And, MB = AB are equal since they are sides of the square ABMN.Now,

∠MBC=∠ABD
  since they both are of measure

90°+∠ABC
  each.Hence, by using SAS congruence theorem,

ΔMBC≅ΔABD
 .And,

area(MBC)=area(ABD)
 .                                   ……(1)So,

ΔABD
 and square BYXD have the same base BD and are between the same parallels BD and AX.Thus,

area(ΔABD)=
      1
      2
     
     ×area(BYXD)

⇒area(BYXD)=2×area(ΔABD)

By (1),

⇒ar(BYXD)=2ar(ΔMBC)
                                        ……(2)Square ABMN and

ΔMBC
  have the same base and are between the same parallels MB and NC.

2area(ΔMBC)=area(square ABMN)
                          ……(3)By combining (2), (3),

⇒area(BYXD)=area(ABMN)
 Hence, the statement is true.Therefore, option 1 is correct.

Detailed Solution