Question

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​

Answer 1

✬ 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 ★

$\implies{\rm }$ AC² = AD² + DC²

$\implies{\rm }$ a² = AD² + (a/2)²

$\implies{\rm }$ a² = AD² + a²/4

$\implies{\rm }$ a² – a²/4 = AD²

$\implies{\rm }$ 4a² – a²/4 = AD²

$\implies{\rm }$ √3a²/4 = AD

$\implies{\rm }$ √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 ★

$\implies{\rm }$ tan60° = AD/DC

$\implies{\rm }$ √3 = AD/a/2

$\implies{\rm }$ √3 × a/2 = AD

$\implies{\rm }$ √3a/2 = AD

Hence, the measure of altitude of equilateral with side a is √3a/2.

Answer 2

Answer:

Given :-

• ABC is an equilateral traingle
• Side = a units

To Find :-

Altitude

Solution :-

Let O be the part of right triangle which divide ABC into 2 equal parts

Now,

Let a perpendicular bisector AO on BC in which,

• AO ⟂ BC

Now,

Since O is dividing into both parts,

So,

∠AOC = ∠AOB = 90°

Now,

BO = OC = BC/2

BO = OC = a/2

By using Pythagoras Theorem

H² = P² + B²

H is the Hypotenuse

P is the Perpendicular

B is the Base

AC² = AO² + OC²

Now,

• AC = a
• AO = AO
• OC = a/2

a² = AO² + a/2²

a² = AO² + a × a/2 × 2

a² = AO² + a²/4

AO² = a² - a²/4

AO² = 4a² - a²/4

AO² = 3a²/4

AO² = √3a/2

$\large \sf \: AO = \sqrt{3}\bigg(\dfrac{a}{2}\bigg)$