Problem 51 - Elementary Algebra Within Reach

Question

Geometry The sides of a yield sign all have the same length (see figure). The perimeter of a roadway yield sign is 225 centimeters. Find the length of each side.

Step-by-Step Solution

Answer

The length of each side of the yield sign is 75 centimeters.

Step 1: Understand the Geometry of the Yield Sign

A yield sign is commonly represented as a triangle, specifically an equilateral triangle. In an equilateral triangle, all sides have the same length.

Step 2: Calculate the Length of Each Side

Knowing that the perimeter is 225 centimeters and that a triangle has 3 sides, divide the total perimeter by 3, the number of sides, to find the length of each side. Use the formula for length of a side \( length = \frac{Perimeter}{Number of Sides} \)

Step 3: Perform the Calculation

By plugging the values from the problem into the formula from step 2, the length of each side of the yield sign can be calculated as \( length = \frac{225cm}{3}\).

Key Concepts

Geometry and the Equilateral Triangle

Geometry is a branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects and spaces. When we dive into the world of geometric shapes, the equilateral triangle stands out for its distinct properties. An equilateral triangle is a polygon with three straight sides, where each side is exactly the same length, and each of the three angles is exactly 60 degrees.

An understanding of this uniformity becomes the foundation for solving problems related to these triangular shapes, such as finding the perimeter, which is the total length around the triangle. The properties of equilateral triangles are fundamental in geometry because they simplify many equations due to their symmetrical nature. This symmetry also lends to their frequent appearance in man-made designs, such as the roadway yield sign mentioned in our exercise.

Calculating the Perimeter of Shapes

The perimeter of a shape in geometry refers to the total length of its boundaries. For polygons, which are shapes with straight sides, finding the perimeter entails summing the lengths of all sides. The formula to calculate the perimeter varies based on the polygon type.

For an equilateral triangle, with all sides equal, the perimeter calculation is particularly straightforward. It involves multiplying the length of one side by the total number of sides, which is three for a triangle. In our yield sign example, the equation simplifies to multiplying the length of one side by three. The perimeter can provide crucial information, such as the amount of material needed to create a frame around a shape, which is directly applicable to various real-world scenarios.

Equilateral Triangle Properties

An equilateral triangle is not just a figure with three equal sides; it's a study in symmetry. This unique shape has all three interior angles measuring 60 degrees. These equal angles contribute to the triangle's ability to tessellate, or fit together with exact replicas, without gaps or overlaps.

The properties of the equilateral triangle result in simplifications in geometry. For example, since all sides are of the same length, finding the length of one side instantly answers the lengths of the others, streamlining calculations such as the perimeter. This uniformity also means that the triangle's centroid, incenter, circumcenter, and orthocenter coincide at a single point, which is not the case for other triangle types. By fully understanding these properties, students can easily grasp more complex geometric concepts.