Calculating the Height of an Equilateral Triangle

The question asks us to find the height of an equilateral triangle given its side length. An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal (each being 60 degrees).

Given Information

• Side length of the equilateral triangle, \(s = 4\sqrt{3}\) cm.

Finding the Height of an Equilateral Triangle

There is a standard formula to find the height (\(h\)) of an equilateral triangle when its side length (\(s\)) is known. The formula is derived using the Pythagorean theorem, considering half of the equilateral triangle which forms a 30-60-90 right-angled triangle.

The formula for the height of an equilateral triangle is:

\[ h = \frac{\sqrt{3}}{2} s \]

Step-by-Step Calculation

Now, we will substitute the given side length \(s = 4\sqrt{3}\) cm into the formula:

\[ h = \frac{\sqrt{3}}{2} \times (4\sqrt{3}) \]

Multiply the terms:

\[ h = \frac{\sqrt{3} \times 4 \times \sqrt{3}}{2} \]

Recall that \(\sqrt{a} \times \sqrt{a} = a\). So, \(\sqrt{3} \times \sqrt{3} = 3\).

\[ h = \frac{4 \times (\sqrt{3} \times \sqrt{3})}{2} \]

\[ h = \frac{4 \times 3}{2} \]

\[ h = \frac{12}{2} \]

Divide 12 by 2:

\[ h = 6 \]

So, the height of the equilateral triangle is 6 cm.

Comparing with Options

Let's look at the given options:

• Option 1: 4 cm
• Option 2: \(6\sqrt{3}\) cm
• Option 3: \(4\sqrt{3}\) cm
• Option 4: 6 cm

Our calculated height is 6 cm, which matches Option 4.

Summary of Calculation

Revision Table: Equilateral Triangle Formulas

Additional Information: Understanding Equilateral Triangles

An equilateral triangle is a special type of triangle that possesses several unique properties:

• Equal Sides: All three sides are of the same length.
• Equal Angles: All three interior angles are equal, each measuring 60 degrees. This makes it an acute triangle.
• Symmetry: An equilateral triangle has three lines of symmetry, rotational symmetry of order 3, and reflectional symmetry.
• Altitude, Median, Angle Bisector, Perpendicular Bisector: In an equilateral triangle, the altitude from any vertex to the opposite side is also the median to that side, the angle bisector of the vertex angle, and the perpendicular bisector of the opposite side. This altitude is what we calculate as the 'height'.
• Relationship between Height and Side: The height divides the equilateral triangle into two congruent 30-60-90 right-angled triangles. In a 30-60-90 triangle, the sides are in the ratio \(x : x\sqrt{3} : 2x\), where \(x\) is the side opposite the 30-degree angle. For an equilateral triangle with side \(s\), the base of the right triangle is \(s/2\), the height is \(h\), and the hypotenuse is \(s\). Using the ratio, \(s/2\) corresponds to \(x\), \(h\) corresponds to \(x\sqrt{3}\), and \(s\) corresponds to \(2x\). From \(s = 2x\), we get \(x = s/2\). Substituting into \(h = x\sqrt{3}\), we get \(h = (s/2)\sqrt{3} = \frac{\sqrt{3}}{2}s\). This confirms the height formula used in the solution.