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Find the length of the side and perimeter of an equilateral triangle whose height is \[\sqrt{3}\] cm

Chapter 2 – Pythagoras Theorem- Text Book Solution

Problem Set 2 | Q 5 | Page 44

solution

Since, ABC is an equilateral triangle, CD is the perpendicular bisector of AB.
Now, According to Pythagoras theorem,
In ∆ACD

\[{AC}^2 = {AD}^2 + {CD}^2 \]
\[ \Rightarrow \left( 2a \right)^2 = a^2 + \left( \sqrt{3} \right)^2 \]
\[ \Rightarrow 4 a^2 - a^2 = 3\]
\[ \Rightarrow 3 a^2 = 3\]
\[ \Rightarrow a^2 = 1\]
\[ \Rightarrow a = \sqrt1 = 1\] cm

Hence, the length of the side of an equilateral triangle is 2 cm.
Now,
Perimeter of the triangle = (2 + 2 + 2) cm
= 6 cm
Hence, perimeter of an equilateral triangle is 6 cm.

Explanation:-

In an equilateral triangle, all sides are equal and all angles are 60 degrees. The height of an equilateral triangle bisects one of the sides and forms a right triangle with half of that side.

Let’s denote the length of one side of the equilateral triangle as “s”. Then, the length of the base of the right triangle formed by the height is s/2. Using the Pythagorean theorem, we can find the length of the other leg of the right triangle:

(s/2)^2 + h^2 = s^2 s^2/4 + 3 = s^2 3s^2/4 = 3 s^2 = 4 s = 2 cm

So, the length of each side of the equilateral triangle is 2 cm. The perimeter of an equilateral triangle is simply three times the length of one side:

perimeter = 3s perimeter = 3 x 2 cm perimeter = 6 cm

Therefore, the length of the side of the equilateral triangle is 2 cm and the perimeter is 6 cm.

Chapter 2 – Pythagoras Theorem- Text Book Solution

Problem Set 2 | Q 5 | Page 44