The perimeter of an equilateral triangle is 60 cm. Find its (i) area and (ii) height. (Given, `sqrt(3) = 1732`.)

Options

A. `17.32` cm
B. `12.32` cm
C. `14.32` cm
D. `16.32` cm

Explanation

To solve the problem, we will follow these steps:

### Step 1: Find the length of each side of the equilateral triangle.
The perimeter of an equilateral triangle is given as 60 cm. Since all three sides are equal, we can find the length of one side (a) by dividing the perimeter by 3.

\[ a = \frac{\text{Perimeter}}{3} = \frac{60 \, \text{cm}}{3} = 20 \, \text{cm} \]

### Step 2: Calculate the area of the equilateral triangle.
The formula for the area (A) of an equilateral triangle is given by:

\[ A = \frac{\sqrt{3}}{4} a^2 \]

Substituting the value of \(a\):

\[ A = \frac{\sqrt{3}}{4} (20 \, \text{cm})^2 = \frac{\sqrt{3}}{4} \times 400 \, \text{cm}^2 = 100\sqrt{3} \, \text{cm}^2 \]

Now, using the approximation \(\sqrt{3} \approx 1.732\):

\[ A \approx 100 \times 1.732 = 173.2 \, \text{cm}^2 \]

### Step 3: Calculate the height of the equilateral triangle.
The formula for the area can also be expressed in terms of the base and height:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, the base is equal to the length of one side (20 cm). Let \(h\) be the height. Thus, we have:

\[ 173.2 = \frac{1}{2} \times 20 \times h \]

Solving for \(h\):

\[ 173.2 = 10h \]

\[ h = \frac{173.2}{10} = 17.32 \, \text{cm} \]

### Final Answers:
(i) Area of the triangle = 173.2 cm²
(ii) Height of the triangle = 17.32 cm

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