State true or false.In the following figure,

ABC
and

BDE
and are two equilateral triangles such that

D
are two equilateral triangles such that is the mid-point of

BC
is the mid-point of .

AE
. intersects

BC
intersects at

F
at . Then,

ar

ΔBDE
=
1
2

ar

ΔBAE

. Then, .

Question

State true or false.In the following figure,

ABC
  and

BDE
  and  are two equilateral triangles such that

D
  are two equilateral triangles such that  is the mid-point of

BC
  is the mid-point of .

AE
 .  intersects

BC
  intersects  at

F
  at . Then,

ar
    
     ΔBDE
   = 
    1
    2
   
   ar
    
     ΔBAE
   
 . Then, .

Accepted Answer

Given that,

ABC
  and

BDE
  are two equilateral triangles such that

D
  is the mid-point of

BC
  and

AE
  intersects

BC
  in

F
 .We have to prove that,

ar
    
     Δ BDE
   = 
    1
    2
   
   ar
    
     Δ BAE
   
 .Since DE is the median of

ΔBEC
  and we know that the median divides the triangle into two triangles of equal areas.

ar(ΔBDE)=
    1
    2
   
   ar(ΔBEC)
                             ……(1)

∠EBC=∠BCA=60°
 This suggest that they are alternate  angles between the lines BE and AC with BC as the transversal.So,

BE∥AC
 .Since triangles

BAE and BEC
  are on the same base

BE
  and between the same parallels

BE and AC
 . Therefore, area

(ΔBAE)
 = area

(ΔBCE)
 .     …..(1)Calculate area

(ΔBDE)
 . From (1),area

(ΔBEC)
  = 2 area

(ΔBDE)
          ….(2)Area

(ΔBDE)
  =

1
    2

area

(ΔBAE)
 Hence, the statement is true.Hence, option 1 is correct.

State true or false.

In the following figure, $A B C$ and $B D E$ and are two equilateral triangles such that $D$ are two equilateral triangles such that is the mid-point of $B C$ is the mid-point of . $A E$ . intersects $B C$ intersects at $F$ at . Then, $a r Δ B D E = 12 a r Δ B A E$ . Then, .

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App

State true or false.In the following figure, ABC and BDE and are two equilateral triangles such that D are two equilateral triangles such that is the mid-point of BC is the mid-point of . AE . intersects BC intersects at F at . Then, ar ΔBDE = 1 2 ar ΔBAE . Then, .