∆ABC and ∆DEF are equilateral triangles.If A(∆ABC):A(∆DEF)=1:2 and AB=4,then what is the length of DE?
Answers
Answer:
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Answer:
The length of DE is 4 √2 units.
Step-by-step-explanation:
We have given that,
△ABC and △DEF are equilateral triangles.
∴ ∠A = ∠B = ∠C = ∠D = ∠E = ∠F = 60° - - [ Angles of equilateral triangle ]
Now,
In △ABC and △DEF,
∠A ≅ ∠D - - [ Each of 60°]
∠B ≅ ∠E - - [ Each of 60°]
∴ △ABC ∼ △DEF - - [ AA test of similarity ]
Now,
A ( △ABC ) / A ( △DEF ) = AB² / DE² - - ( 1 ) [ Ratio of areas of similar triangles ]
A ( △ABC ) : A ( △DEF ) = 1 : 2 - - ( 2 ) [ Given ]
From ( 1 ) & ( 2 )
⇒ 1 / 2 = 4² / DE²
⇒ 1 / 2 = 16 / DE²
⇒ DE² = 16 × 2
⇒ DE = 4 √2 - - [ Taking square roots ]
∴ The length of DE is 4 √2 units.
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Additional Information:
1. Similarity:
When two things are same in the view, then they are similar.
The best example of similarity is we and our photo. Both are same in view, but different in measurements.
2. Similarity in triangles:
In two triangles, when corresponding angles are congruent and corresponding sides are in proportion, the two triangles are known as similar.
3. Ratio of area of two triangles:
When two triangles are similar, the ratio of their area is equal to the ratio of squares of the corresponding sides.