Question

Triangle ABC and triangle BDE are a equilateral triangles such that D is the mid point of BC Ratio of the areas of a similar triangles are
a)2:1
b) 1:2
c)1:4
d)4:1
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Answer 1

Answer

Option d) 4 : 1

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Given

∆ABC and ∆BDE are equilateral triangle. D is the midpoint of BC (check attachment for figure).

To Find

Ratio of area of the two similar triangle.

Solution

Since, D is the midpoint of BC,

→ BD = BC/2

→ BC = 2BD

We know that equilateral triangles have all angles equal to 60°. Hence, all equilateral triangles are similar to each other.

Hence, here we can write

BC/BD = AB/BE = AC/DE

BC/BD = 2BD/BD (Since, BC = 2BD)

→ BC/BD = 2/1

Now, we know that ratio of area of two similar triangles is equal to the square of ratio of the corresponding sides.

→ ratio of area of two triangles = (BC/BD)²

= (2/1)²

= 4/1

Hence, ratio of ar(ABC) : ar(BED) is 4 : 1.

Answer 2

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In Δ ABC and Δ BDE

∠ABC = ∠ABC (common)

∠BAC = ∠BED (60° each)

⟹ ΔABC ∼ Δ BED

======================>

As E is mid point :-

⟹ BC = 2BD

⟹ BA = 2AE

========>

BC/BD = AB/BE = AC/DE

BC/BD = 2BD/BD

BC/BD = 2/1

==================>

Now take areas

(BC/BD)²

= (2/1)²

= 4/1