Q1.5.
If ABC is an equilateral triangle of side a, then its altitude is equal to
√за
з
(2)
(3) За
(4)
3
5
а
(1)
a
2
Answers
Explanation:
✬ Altitude = √3a/2 ✬
Step-by-step explanation:
Given:
Measure of side of an equilateral triangle is 'a'.
To Find:
What is measure of its altitude ?
Solution: Let ∆ABC be an equilateral triangle where
AB = BC = CA = a
Construction: Draw a perpendicular bisector AD on BC such that
AD ⟂ BC
∠ADC = ∠ADB = 90°
BD = DC { perpendicular bisects the side BC in two equal halves }
➼ BD = DC = BC/2
➼ BD = DC = a/2
Now , in right angled ∆ADC by using Pythagoras Theorem -
AD (perpendicular)
DC (base)
AC (hypotenuse)
★ H² = Perpendicular² + Base² ★
⟹
⟹ AC² = AD² + DC²
⟹ a² = AD² + (a/2)²
⟹ a² = AD² + a²/4
⟹ a² – a²/4 = AD²
⟹ 4a² – a²/4 = AD²
⟹ √3a²/4 = AD
⟹ √3a/2= AD
Hence, the measure of altitude of equilateral with side a is √3a/2.
______________________
[ Solving through an another method ]
AC (hypotenuse) = a
DC (base) = a/2
AD (perpendicular)
As we know that
➮ Measure of each side of an equilateral triangle is 60°.
So in ∆ADC , using tanθ
★ tanθ = Perpendicular/Base ★
⟹ tan60° = AD/DC
⟹ √3 = AD/a/2
⟹ √3 × a/2 = AD
⟹ √3a/2 = AD
Hence, the measure of altitude of equilateral with side a is √3a/2.