Question

Geometry An equilateral triangle has a height of 3 feet. How long are each of its legs? (Hint: Use the height of the triangle to partition the triangle into two right triangles of the same size.)

Short answer

The length of each side of the equilateral triangle is \(2\sqrt{3}\) feet.

Step-by-step solution

Derive two right-angled triangles from the equilateral triangle

By drawing a line from one peak to the base, we can partition the given equilateral triangle into two right-angled triangles. The line drawn now serves as the height of the equilateral triangle, which is 3 feet.

Find the length of the base

The line drawn from the peak to the base bisects the base into two equal parts. So each part of the base is half the length of one of the sides of the equilateral triangle. Let this side be 'x' feet. Thus, each part of the base is \(x/2\) feet.

Use Pythagorean theorem to derive a relation between height and base of right triangles

For a right triangle, the Pythagorean theorem states that the square of the hypotenuse (long side) is equal to the sum of the squares of the other two sides. Therefore, \(x^2 = h^2 + b^2\), where x is the length of the hypotenuse, h is the height and b is the base.

Substitute values

We know from previous steps that h = 3 feet and b = \(x/2\). Substituting these values in the Pythagorean theorem, we get \(x^2 = 3^2 + (x/2)^2\)

Simplify the equation

The equation simplifies to \(x^2 = 9 + x^2/4\). Multiply entire equation by 4 to get rid of the division, we get \(4x^2 = 36 + x^2\). Simplifying further gives \(3x^2 = 36 => x^2 = 12 => x = \sqrt{12}\)

Find the length of x

...which after taking the square root, we get \(x = 2\sqrt{3}\) feet

Key concepts

Right Triangle

A right triangle is a triangle where one of the angles is exactly 90 degrees. This type of triangle is very important in geometry because it forms a foundation for many concepts, especially when working with angles and dimensions. In the original exercise, the height of the equilateral triangle forms a right triangle when it splits the equilateral triangle into two equal right triangles.

Each of these right triangles shares:

• The original height of the equilateral triangle
• Half the base of the equilateral triangle
• The hypotenuse, which is equivalent to one side of the equilateral triangle

Recognizing right triangles is a key strategy, as it allows us to apply specific mathematical rules, the most famous being the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

In the context of partitioning the equilateral triangle from the original exercise, the Pythagorean theorem helps determine the length of the sides. By knowing the height (3 feet) and half the base (\( x/2 \)), the theorem allows us to calculate the length of the side of the equilateral triangle, which is the hypotenuse in this scenario. Applying the theorem simplifies problem-solving by converting complex problems into manageable steps using consistent rules.

Geometric Partitioning

Geometric partitioning refers to breaking down a complex shape into simpler parts, making it easier to analyze and understand. In geometry, especially when dealing with triangles, partitioning helps by reducing a problem into smaller, more solvable components.

For an equilateral triangle, drawing a line from one vertex to the midpoint of the opposite side effectively divides the triangle into two right triangles. This partitioning is particularly useful because it converts a non-right shape into familiar right triangles where tools like the Pythagorean theorem can be used conveniently.

• This approach simplifies calculations
• Allows visualization of the problem
• Makes use of symmetrical properties

Using geometric partitioning not only aids in visualization but also provides an easier path to solving the problem by reducing it to repeatable, smaller processes.