The Base Of A Solid Oblique Pyramid Is An Equilateral Triangle With An Edge Length Of S S S Units. Which Expression Represents The Height Of The Triangular Base Of The Pyramid?A. 5 2 2 \frac{5}{2} \sqrt{2} 2 5 ​ 2 ​ Units B. $\frac{5}{2}

Introduction

In geometry, a pyramid is a three-dimensional shape with a base and sides that converge at the apex. The base of a solid oblique pyramid is an equilateral triangle with an edge length of $s$ units. In this article, we will explore the concept of the height of the triangular base of the pyramid and derive an expression to represent it.

Understanding the Triangular Base

The triangular base of the pyramid is an equilateral triangle, which means all its sides are equal in length. The edge length of the triangle is given as $s$ units. To find the height of the triangular base, we need to understand the properties of an equilateral triangle.

Properties of an Equilateral Triangle

An equilateral triangle has several properties that are essential in finding its height. Some of these properties include:

• All sides are equal: In an equilateral triangle, all three sides are equal in length. In this case, the edge length of the triangle is $s$ units.
• All angles are equal: The angles of an equilateral triangle are all equal, and each angle measures 60 degrees.
• Altitude divides the triangle into two 30-60-90 triangles: When an altitude is drawn from a vertex of an equilateral triangle to the opposite side, it divides the triangle into two 30-60-90 triangles.

Finding the Height of the Triangular Base

To find the height of the triangular base, we can use the properties of an equilateral triangle. Since the altitude divides the triangle into two 30-60-90 triangles, we can use the ratios of the sides of a 30-60-90 triangle to find the height.

Step 1: Draw the Altitude

Draw an altitude from one of the vertices of the equilateral triangle to the opposite side. This altitude divides the triangle into two 30-60-90 triangles.

Step 2: Identify the Ratios of the Sides

In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle.

Step 3: Find the Height

Since the altitude is the side opposite the 60-degree angle, we can use the ratio of the sides to find its length. The length of the altitude is √3 times the length of the side opposite the 30-degree angle, which is half the length of the hypotenuse.

Step 4: Express the Height in Terms of s

The length of the hypotenuse is equal to the edge length of the equilateral triangle, which is $s$ units. Therefore, the length of the altitude is:

$$\frac{\sqrt{3}}{2} s$$

Conclusion

In conclusion, the height of the triangular base of the pyramid is $\frac{\sqrt{3}}{2} s$ units. This expression represents the height of the triangular base in terms of the edge length of the equilateral triangle.

Discussion

The discussion category for this article is mathematics. The article explores the concept of the height of the triangular base of a pyramid and derives an expression to represent it. The expression is $\frac{\sqrt{3}}{2} s$ units, where $s$ is the edge length of the equilateral triangle.

References

• [1] Geometry: A Comprehensive Introduction
• [2] Mathematics for Dummies

Final Answer

Q&A: Frequently Asked Questions

Q: What is the height of the triangular base of a pyramid?

A: The height of the triangular base of a pyramid is $\frac{\sqrt{3}}{2} s$ units, where $s$ is the edge length of the equilateral triangle.

Q: Why is the height of the triangular base important?

A: The height of the triangular base is important because it is a key component in determining the volume of the pyramid. The volume of a pyramid is given by the formula $\frac{1}{3} \times \text{base area} \times \text{height}$.

Q: How do you find the height of the triangular base?

A: To find the height of the triangular base, you can use the properties of an equilateral triangle. Since the altitude divides the triangle into two 30-60-90 triangles, you can use the ratios of the sides of a 30-60-90 triangle to find the height.

Q: What are the properties of an equilateral triangle?

A: An equilateral triangle has several properties that are essential in finding its height. Some of these properties include:

• All sides are equal: In an equilateral triangle, all three sides are equal in length.
• All angles are equal: The angles of an equilateral triangle are all equal, and each angle measures 60 degrees.
• Altitude divides the triangle into two 30-60-90 triangles: When an altitude is drawn from a vertex of an equilateral triangle to the opposite side, it divides the triangle into two 30-60-90 triangles.

Q: What is a 30-60-90 triangle?

A: A 30-60-90 triangle is a right triangle with angles measuring 30, 60, and 90 degrees. The ratio of the sides of a 30-60-90 triangle is 1:√3:2.

Q: How do you use the ratios of the sides of a 30-60-90 triangle to find the height?

A: To find the height of the triangular base, you can use the ratios of the sides of a 30-60-90 triangle. The length of the altitude is √3 times the length of the side opposite the 30-degree angle, which is half the length of the hypotenuse.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{\sqrt{3}}{2} s}$.

Conclusion

In conclusion, the height of the triangular base of a pyramid is $\frac{\sqrt{3}}{2} s$ units, where $s$ is the edge length of the equilateral triangle. This expression represents the height of the triangular base in terms of the edge length of the equilateral triangle.

Discussion

The discussion category for this article is mathematics. The article explores the concept of the height of the triangular base of a pyramid and derives an expression to represent it. The expression is $\frac{\sqrt{3}}{2} s$ units, where $s$ is the edge length of the equilateral triangle.

References

• [1] Geometry: A Comprehensive Introduction
• [2] Mathematics for Dummies

Final Answer

The final answer is $\boxed{\frac{\sqrt{3}}{2} s}$.