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Q.


What is the altitude of an equilateral triangle in terms of its side length ‘a’?

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a

a 2  units    

b

a 3   units

c

2 a 3   units

d

3 a 2   units       

answer is D.

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Detailed Solution

Given that the side of an equilateral triangle is ‘a’ units.
Let us consider an equilateral triangle ABC with side ‘a’.
Draw a perpendicular from A to BC which intersects base BC at D.
 Question ImageWe can observe from the figure that,
⇒∠ADB=∠ADC= 90 ∘    ------(1)
Now in ΔADB   and ΔADC  ,
AD = AD  [Common]
AB = AC  [Sides of equilateral Triangle]
∠ADB=∠ADC   [From (1)]
By RHS congruence criteria, ΔADB≅ΔADC  .
So, BD = DC.
BD+DC=a ⇒BD+BD=a ⇒2BD=a ⇒BD=DC= a 2  
Pythagoras theorem states that in a right-angled triangle, (hypotenuse) 2 = (base) 2 + (perpendicular) 2  .
Now in ΔADB  , using Pythagoras theorem, we have
⇒A D 2 +B D 2 =A B 2  
Now, substitute AB = a, BD= a 2  .
⇒A D 2 + a 2 = a 2 2 ⇒A D 2 = a 2 − a 2 2 ⇒AD= 4 a 2 − a 2 4 ⇒AD= a 4−1 2 ⇒AD= 3 2 a   
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What is the altitude of an equilateral triangle in terms of its side length ‘a’?