Problem 45 - Algebra 2

Question

window is in the shape of an equilateral triangle. Each side of the triangle is 8 feet long. The window is divided in half by a support from one vertex to the midpoint of the side of the triangle opposite the vertex. Approximately how long is the support? F. 5.7 ft G. 6.9 ft H. 11.3 ft J. 13.9 ft

Step-by-Step Solution

Answer

The support is approximately 6.9 ft long (option G).

Step 1: Identify Triangle Properties

The window is an equilateral triangle, meaning all sides are equal in length and all angles are 60 degrees. Each side of this triangle is given as 8 feet.

Step 2: Determine Perpendicular Support Position

The support divides the equilateral triangle into two congruent 30-60-90 right triangles. The support is the altitude of the equilateral triangle, drawn from one vertex to the midpoint of the opposite side.

Step 3: Apply 30-60-90 Triangle Rule

In a 30-60-90 triangle, the ratio of the shortest side (opposite the 30-degree angle) to the hypotenuse (opposite the 90-degree angle) to the altitude (opposite the 60-degree angle) is 1:2:\(\sqrt{3}\). The base of each right triangle formed is half of one side of the equilateral triangle, i.e., 4 feet.

Step 4: Calculate the Altitude

Using the ratio and given half-base, the formula for the altitude is \(altitude = \text{{base}} \times \sqrt{3}\). Substitute the base of 4 feet to get \( altitude = 4 \times \sqrt{3} \approx 4 \times 1.732 = 6.928\).

Step 5: Verify and Choose the Answer

The altitude, which represents the length of the support, is approximately 6.9 feet. Therefore, the correct answer is G: 6.9 ft.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are the same length. Each angle in an equilateral triangle measures 60 degrees, adding up to a total of 180 degrees for the entire triangle. This equality in both sides and angles gives the triangle its name "equilateral," meaning "equal sides."

Here are some important properties of equilateral triangles:

• All interior angles are congruent and equal 60 degrees.
• Equilateral triangles are also equiangular, meaning all angles are equal.
• The same-length sides and equal angles make these triangles the most symmetric of all triangle types.

Understanding these properties is essential because they allow us to use specific mathematical rules and ratios to solve problems effectively, such as calculating the altitude in a 30-60-90 triangle derived from an equilateral triangle.

Geometric Properties

The geometric properties of equilateral triangles make them fascinating and predictable. In this case, when we divide an equilateral triangle with a line from a vertex to the midpoint of the opposite side, we create two smaller triangles known as 30-60-90 right triangles.

Here’s why 30-60-90 triangles are significant:

• It's a type of special right triangle where the angles are 30, 60, and 90 degrees.
• The side lengths follow a specific ratio of 1: \(2\): \(\sqrt{3}\). This refers to the shortest side (opposite the 30-degree angle), the hypotenuse (double the shortest side), and the altitude (opposite the 60-degree angle).

These properties are crucial because mechanical applications often involve such divisions for supports and measurements, like calculating the altitude of an equilateral triangle used in a window.

Altitude Calculation

Calculating the altitude of an equilateral triangle is a common mathematical task. For an equilateral triangle, the altitude (or height) splits the triangle into two congruent 30-60-90 triangles, each with specific side length ratios.

To calculate the altitude in a 30-60-90 triangle:

• The altitude corresponds to the length opposite the 60-degree angle and is calculated using the ratio \(\sqrt{3}\) times the shorter side (opposite the 30-degree angle).
• In our example, if the whole side length of the equilateral triangle is 8 feet, the shorter side of the 30-60-90 triangle is half, i.e., 4 feet.
• The formula for the altitude is \(4 \times \sqrt{3}\), which simplifies to \(4 \times 1.732\approx 6.928\).

Therefore, the calculated altitude, or the support's length in our example, accurately measures approximately 6.9 feet, illustrating the precise nature of math when applied to geometric properties.