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

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a 2

units

a 3

units

2 a 3

units

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 Image

We can observe from the figure that,

⇒ ∠ A D B = ∠ A D C = 90 ∘

------(1)
Now in

Δ A D B

and

Δ A D C

,
AD = AD [Common]
AB = AC [Sides of equilateral Triangle]

∠ A D B = ∠ A D C

[From (1)]
By RHS congruence criteria,

Δ A D B ≅ Δ A D C

.
So, BD = DC.

B D + D C = a ⇒ B D + B D = a ⇒ 2 B D = a ⇒ B D = D C = a 2

Pythagoras theorem states that in a right-angled triangle,

(h y p o t e n u s e) 2 = (b a s e) 2 + (p e r p e n d i c u l a r) 2

.
Now in

Δ A D B

, using Pythagoras theorem, we have

⇒ A D 2 + B D 2 = A B 2

Now, substitute AB = a,

B D = a 2

⇒ A D 2 + a 2 = a 2 2 ⇒ A D 2 = a 2 − a 2 2 ⇒ A D = 4 a 2 − a 2 4 ⇒ A D = a 4 − 1 2 ⇒ A D = 32 a

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