Question

The altitude of an equilateral triangle of side x cm is

Answer 1

Answer:

Step-by-step explanation:

ket us take altitude x =  2root 3

Altitude of a equilateral triangle=

√3/2 × side

Altitude=√3/2 ×2√3 = 3 cm

Answer 2

Answer:

The altitude of the equilateral triangle  = $\frac{\sqrt{3}x }{2}$cm,  where the length of the side = cm

Step-by-step explanation:

Recall the concepts,

In an equilateral triangle, the median and altitude are the same line segments.

Pythagoras Theorem,

In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Solution:

Given, side of an equilateral triangle = xcm

AB = BC = AC = xcm------------(1)

Let ΔABC be an equilateral triangle. Let AD be the altitude from vertex A to BC.

Required to find, the length of AD

Since ΔABC is an equilateral triangle, AD is also the median of the  ΔABC, and BD = DC = $\frac{1}{2}$BC = $\frac{x}{2}$ -------------(2)

Since  AD is the altitude of the ΔABC, we have ΔABD as an equilateral triangle with right angle at D

Then by Pythagoras theorem, we have

AB² = AD² + BD²

Substituting the values of AB and BD from equations (1) and (2) we get

x² = AD² + $(\frac{x}{2}) ^2$

AD² = x² - $(\frac{x}{2}) ^2$ = x² - $\frac{x^2}{4}$ = $\frac{4x^2- x^2}{4}$ = $\frac{3x^2}{4}$

AD = $\sqrt{\frac{3x^2}{4} }$

AD = $\frac{\sqrt{3}x }{2}$

∴ The altitude of the equilateral triangle  = $\frac{\sqrt{3}x }{2}$cm,  where length of the side = xcm

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