Question

The length of one side of an equilateral triangle is 10 meters. a. Find the length of an altitude. b. Find the area of the triangle.

Short answer

The altitude is \(5\sqrt{3}\) meters, and the area is \(25\sqrt{3}\) square meters.

Step-by-step solution

Understanding the Problem

An equilateral triangle has all sides of equal length, and we need to find the length of its altitude and its area given that each side is 10 meters.

Formula for Altitude of an Equilateral Triangle

Use the formula for the altitude \( h \) of an equilateral triangle, which is given by \( h = \frac{\sqrt{3}}{2} s \), where \( s \) is the length of a side.

Calculate the Altitude

Substitute \( s = 10 \) meters into the formula. So, \( h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \) meters.

Formula for Area of an Equilateral Triangle

Use the formula for the area \( A \) of an equilateral triangle, which is \( A = \frac{\sqrt{3}}{4} s^2 \).

Calculate the Area

Substitute \( s = 10 \) meters into the formula. So, the area \( A = \frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3} \) square meters.

Key concepts

Altitude of Triangle

An equilateral triangle is unique because all its sides and angles are equal. The altitude of a triangle is a perpendicular line from one vertex to the opposite side. For an equilateral triangle, this altitude plays a crucial role in determining many properties.
The formula used to find the altitude "h" of an equilateral triangle is:

• \( h = \frac{\sqrt{3}}{2} s \)

Here, "s" represents the length of a side of the triangle.
So, when given the side length, such as 10 meters, you substitute it into the formula. This gives:

• \( h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \) meters

Thus, the altitude helps to split the triangle into two 30-60-90 right triangles, which is a fundamental concept in geometry, offering insight into other mathematical constructs.

Area of Triangle

Understanding how the area of an equilateral triangle is calculated is fundamental in geometry. The area is a measure of the total space enclosed within the triangle’s boundaries. For an equilateral triangle, the area can be easily calculated using a specific formula.
The formula for the area "A" of an equilateral triangle is:

• \( A = \frac{\sqrt{3}}{4} s^2 \)

"s" again is the side length of the triangle. To find the area when one side is 10 meters, substitute into the formula:

• \( A = \frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3} \) square meters

This formula emerges from considering the triangle's symmetry and using trigonometric identities to resolve height into the base dimensions. Knowing this area formula not only applies to theoretical problems but also to real-world applications where precise measurements are needed.

Geometry Formulas

In geometry, formulas are essential as they capture the relationships between different geometric properties. Formulas for the altitude and area simplify problem-solving for equilateral triangles. Let's summarize the crucial geometry formulas used here:

• Altitude Formula: \( h = \frac{\sqrt{3}}{2} s \) - This helps find the altitude using a triangle's side.
• Area Formula: \( A = \frac{\sqrt{3}}{4} s^2 \) - Utilized to calculate the interior surface area.

These formulas stem from the symmetrical properties of equilateral triangles and are derived using basic geometric principles. Understanding and remembering these tools allows students to efficiently solve problems involving equilateral triangles and also supports more advanced study in geometry.