Problem 39 - Precalculus

Question

Find the area of each triangle with measures given. $$a=15, b=15, c=15$$

Step-by-Step Solution

Answer

The area of the triangle is \( \frac{225 \sqrt{3}}{4} \).

Step 1: Identify the Type of Triangle

This problem gives you the three side lengths of a triangle: \( a = 15 \), \( b = 15 \), and \( c = 15 \). Since all three sides are equal, this is an equilateral triangle.

Step 2: Use the Area Formula for an Equilateral Triangle

For an equilateral triangle with side length \( a \), the formula for the area is given by: \[A = \frac{\sqrt{3}}{4} a^2\] where \( a \) is the length of one side.

Step 3: Substitute Side Length into Formula

Substitute \( a = 15 \) into the area formula: \[A = \frac{\sqrt{3}}{4} \times 15^2\] Simplify the expression: \(15^2 = 225\), so \[A = \frac{\sqrt{3}}{4} \times 225\]

Step 4: Simplify the Result

Calculate the area by simplifying: \[A = \frac{\sqrt{3} \times 225}{4} = \frac{225 \sqrt{3}}{4}\]. This represents the area of the equilateral triangle.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. This also means that all three interior angles are equal, each measuring 60 degrees. Here are some interesting properties of equilateral triangles:

• They are a type of isosceles triangle, where not only two but all three sides are the same.
• Every equilateral triangle is also an equiangular triangle, meaning all its angles are the same.
• Because of its equal sides and angles, it has a symmetrical and balanced shape.

Understanding these properties makes it easy to recognize an equilateral triangle and apply specific formulas to find unknown measures, such as its area.

Triangle Area Formula

To find the area of any triangle, there are several formulas available, depending on the known dimensions. However, when it comes to an equilateral triangle, there is a special formula that simplifies the process: \[A = \frac{\sqrt{3}}{4} \times a^2\]where \(a\) is the length of one side. This formula takes advantage of the symmetrical properties of equilateral triangles, utilizing the fact that all sides and angles are the same.
If provided with a different triangle configuration, another commonly used formula is Heron's Formula when you know all three sides, but for equilateral triangles, the specialized formula is far simpler and efficient.

Geometry

Geometry is a vast field of mathematics concerned with shapes, sizes, and properties of space. Triangles, including equilateral triangles, form a fundamental part of geometric study.

• Geometry helps us understand the properties and relations of points, lines, surfaces, and solids.
• It is divided into various branches, one being Euclidean geometry, which studies flat spaces and is founded on concepts defined by Euclid. This includes the study of triangles.
• Understanding triangles in geometry helps in exploring and solving real-world problems involving spatial relationships.

Geometry not only deals with theoretical concepts but is highly practical in numerous fields like engineering, architecture, and art. Knowledge in geometry is essential for developing problem-solving skills and logical reasoning.