Question

ABC and BDE are two equilateral triangles such that D is the point on BC. If BD : DC = 2 : 1, then the ratio of the area of ΔABC to that of the ΔBDE is?!​

Answer 1

Answer:

2:1

Step-by-step explanation:

because BC is bigger than BD

and

Answer 2

Therefore the ratio of the area of ΔABC to that of the ΔBDE is 9:4.

Given:

ΔABC and ΔBDE are two equilateral triangles D is the point on BC.

BD : DC = 2 : 1

To Find:

The ratio of the area of ΔABC to that of the ΔBDE.

Solution:

This problem of Geometry can be simply solved by using the following method.

Given that: BD : DC = 2 : 1

⇒ BD / DC = 2 / 1 ⇒ BD = 2 DC

⇒ BC = BD + DC = 2DC + DC = 3 DC

According to the Similarity of triangles property on area,

If there are 2 similar triangles namely ΔXYZ and ΔPQR, then the ratio of the areas of ΔXYZ and ΔPQR is given by,

⇒ Ar ( ΔXYZ ) / Ar ( ΔPQR ) = ( Any side in ΔXYZ )² / ( Corresponding side in ΔPQR ) { for example XY² / PQ² }

Now in the given question,

⇒ Ar ( ΔABC ) / Ar ( ΔBDE ) = BC² / BD²

⇒ Ar ( ΔABC ) / Ar ( ΔBDE ) = ( 3DC )² / ( 2DC )²       { ∵ BC = 2DC and BD = 2DC }

⇒ Ar ( ΔABC ) / Ar ( ΔBDE ) = 9 / 4

Therefore the ratio of the area of ΔABC to that of the ΔBDE is 9:4.

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