Finding the Altitude of an Equilateral Triangle Given its Perimeter

Understanding the properties of an equilateral triangle is key to solving this problem. An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal (each being 60 degrees).

We are given the perimeter of the equilateral triangle, which is 18 m.

The perimeter of any polygon is the total length of all its sides. For an equilateral triangle with side length 's', the perimeter is the sum of its three equal sides:

\( \text{Perimeter} = \text{side} + \text{side} + \text{side} = 3 \times \text{side} \)

Given the perimeter is 18 m, we can find the side length:

\( 18 \text{ m} = 3 \times \text{side} \)

Dividing the perimeter by 3 gives us the length of one side:

\( \text{side} = \frac{18 \text{ m}}{3} = 6 \text{ m} \)

So, each side of the equilateral triangle is 6 m long.

Now we need to find the altitude of this equilateral triangle. The altitude from any vertex of an equilateral triangle is also the median and the angle bisector to the opposite side. It divides the equilateral triangle into two congruent 30-60-90 right-angled triangles.

Consider one of these right-angled triangles. The hypotenuse is the side of the equilateral triangle (6 m), one leg is half of the base (half of 6 m, which is 3 m), and the other leg is the altitude (h) we want to find.

We can use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) where 'c' is the hypotenuse, and 'a' and 'b' are the legs.

\( (\text{half-base})^2 + (\text{altitude})^2 = (\text{side})^2 \)

\( (3 \text{ m})^2 + h^2 = (6 \text{ m})^2 \)

\( 9 \text{ m}^2 + h^2 = 36 \text{ m}^2 \)

Subtract 9 from both sides:

\( h^2 = 36 \text{ m}^2 - 9 \text{ m}^2 \)

\( h^2 = 27 \text{ m}^2 \)

Take the square root of both sides to find 'h':

\( h = \sqrt{27} \text{ m} \)

We can simplify the square root of 27:

\( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \)

So, the altitude \(h = 3\sqrt{3}\) m.

Alternatively, you can use the direct formula for the altitude of an equilateral triangle with side length 's':

\( \text{Altitude} = \frac{\sqrt{3}}{2} \times \text{side} \)

Using the side length \(s = 6\) m:

\( \text{Altitude} = \frac{\sqrt{3}}{2} \times 6 \text{ m} \)

\( \text{Altitude} = 3\sqrt{3} \text{ m} \)

Both methods yield the same result for the altitude of the equilateral triangle.

Revision Table: Equilateral Triangle Formulas

Here is a quick summary of key formulas for an equilateral triangle with side length 's':

• Perimeter = \(3s\)
• Altitude = \( \frac{\sqrt{3}}{2} s \)
• Area = \( \frac{\sqrt{3}}{4} s^2 \)

Additional Information on Equilateral Triangles

Equilateral triangles are special types of triangles with many unique properties:

• All interior angles are 60 degrees.
• The altitude, median, and angle bisector from any vertex are the same line segment.
• The centroid, orthocenter, circumcenter, and incenter all coincide at a single point.
• An equilateral triangle is a regular polygon with 3 sides.
• It has three lines of symmetry.

Knowing these properties helps in solving geometry problems involving equilateral triangles.