Question

A ABC is an Equilateral triangle. We have BD = EG = DF = DE = EC, then the ratio of the area of the
shaded portion to the area of AABC is
4
7
5
6
(A)
(B)
(C)
(D)
11
9
7
12​

Answer 1

Answer:

The ratio is 7/9

Step-by-step explanation:

Refer the photo

Answer 2

Given:

ABC is an equilateral triangle.

BD= EG= DF= DE= EC

To Find:

The ratio of the area of the shaded portion

Solution:

In ΔABC,

∠A = ∠B = ∠C = 60° [∵ angles in equilateral triangle is 60°]

In ΔBFD and ΔGEC

          ∠1 = ∠2

          ∠3 = ∠4    [ isosceles property]

       ∠1 = ∠2 = 60°    [ ∠B = ∠C = 60°]

       ∠3 = ∠4 = 60°

∠1 + ∠2 + ∠5 = 180°

60° + 60° + ∠5 = 180°

                   ∠5 = 60°

    similarly, ∠6 = 60°

  In ΔBFD and ΔCGE

              FD = GE (given)

               BD = CE(given)

                ∠5 = ∠6 (both are 60°)

so, ΔBFD ≅ CGF (SAS congruency rule)

     Area of ΔBFD = Area of ΔCGF    (I)

 So, area of ΔABC = √3a²/4

       area of ΔBFD = √3(9/3)²/4

                                  √3a²/9×1/4

area of AGEDF= area of ΔABC- area of ΔBFD- area of ΔCGF

                        =√3a²/4 - 2×√3a²×4/9×2

                        = √3a²/4 - √3a²/18

                        = 9√3a²-2√3a²/36

                        = 7√3a²/36

Ratio = 7:9or 7/9