Understanding the Equilateral Triangle Problem

The question asks us to find the height of an equilateral triangle ABC. We are given that D is the midpoint of side BC, and the length of BC is 8 cm.

In an equilateral triangle, all sides are equal in length, and all angles are equal to 60 degrees. Since BC is 8 cm, sides AB and AC are also 8 cm each.

The height of an equilateral triangle is the perpendicular distance from a vertex to the opposite side. When we draw the height from vertex A to side BC, it meets BC at point D, because D is the midpoint of BC. This height (AD) is also the median and angle bisector in an equilateral triangle. The height AD is perpendicular to BC.

Using the Pythagorean Theorem

When the height AD is drawn to the base BC, it divides the equilateral triangle ABC into two congruent right-angled triangles, ADB and ADC.

Consider the right-angled triangle ADB:

• The hypotenuse is AB, which is a side of the equilateral triangle, so AB = 8 cm.
• The base is BD, which is half the length of BC because D is the midpoint of BC. So, BD = BC / 2 = 8 cm / 2 = 4 cm.
• The height is AD, which we need to find. Let's call the height \(h\).

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For triangle ADB:

\(AD^2 + BD^2 = AB^2\)

Substituting the values we know:

\(h^2 + 4^2 = 8^2\)

\(h^2 + 16 = 64\)

Now, we solve for \(h^2\):

\(h^2 = 64 - 16\)

\(h^2 = 48\)

To find \(h\), we take the square root of both sides:

\(h = \sqrt{48}\)

We can simplify \(\sqrt{48}\) by finding the largest perfect square factor of 48. 48 can be written as \(16 \times 3\). The square root of 16 is 4.

\(h = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\)

So, the height of the equilateral triangle is \(4\sqrt{3}\) cm.

Alternative Method: Formula for Height of Equilateral Triangle

The height \(h\) of an equilateral triangle with side length \(a\) can be directly calculated using the formula:

\(h = \frac{\sqrt{3}}{2} a\)

In this problem, the side length \(a = 8\) cm. Substituting this value into the formula:

\(h = \frac{\sqrt{3}}{2} \times 8\)

\(h = 4\sqrt{3}\)

Thus, the height of the equilateral triangle is \(4\sqrt{3}\) cm.

Summary of Steps

1. Identify the properties of the equilateral triangle and the given information (side length = 8 cm).
2. Recognize that the height to the base bisects the base, creating a right-angled triangle.
3. Determine the lengths of the sides of the right-angled triangle (hypotenuse = 8 cm, base = 4 cm).
4. Use the Pythagorean theorem (\(h^2 + base^2 = hypotenuse^2\)) or the direct formula for the height of an equilateral triangle (\(h = \frac{\sqrt{3}}{2} a\)).
5. Calculate the height \(h\).

Revision Table: Key Triangle Concepts

Additional Information on Geometric Shapes and Measurements

Geometry deals with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. Triangles are fundamental geometric shapes. Calculating lengths, areas, and volumes are common tasks.

The Pythagorean theorem (\(a^2 + b^2 = c^2\)) is crucial for solving problems involving right-angled triangles. It relates the lengths of the two legs (\(a\) and \(b\)) to the length of the hypotenuse (\(c\)).

Understanding the special properties of equilateral triangles, such as the relationships between their sides, angles, height, median, and angle bisector, simplifies problem-solving. The height not only gives the vertical dimension but also helps in calculating the area of the triangle using the formula: Area = \(\frac{1}{2} \times base \times height\).