Math New Drill (Level 1) : Geometry(Plane figure) Right hexagon in an equilateral triangle

As shown in a lower figure, the right hexagon DEFGHI is in equilateral triangle ABC. 

Three points D, F, and H are on the side of the equilateral triangle and AD : DB = BF : FC = CH : HA = 2 : 3. 

Answer the following questions. 

(1) Find the ratio of the area of equilateral triangle ABC to the area of the triangle ADH. 

(2) Find the ratio of the area of equilateral triangle ABC to the area of the right hexagon DEFGHI.

Answer

(1) 25 : 6 

(2) 25 : 14

Solution

(1)

As shown in Fig.1, the area of △ABC is assumed to be 5 × 5 = 25,

 and the area of △ADH is assumed to be 3 × 2 = 6.

(2)

△DFH = 25 - 6 × 3 = 7

Right hexagon DEFGHI = △DFH × 2 = 7 × 2 =14 as shown in Fig.2.