What Is The Simplest Form Of The Expression For The Side Length, In Feet?A. 4 50 4 \sqrt{50} 4 50 ​ B. 5 6 5 \sqrt{6} 5 6 ​ C. 10 2 10 \sqrt{2} 10 2 ​ D. 20 2 20 \sqrt{2} 20 2 ​

Introduction

In geometry, the side length of a triangle can be expressed in various forms, including radical expressions. Simplifying these expressions is essential to understand the properties of the triangle and to perform calculations accurately. In this article, we will explore the simplest form of the expression for the side length of a triangle.

Understanding the Problem

The problem presents four options for the simplest form of the expression for the side length of a triangle:

A. $4 \sqrt{50}$ B. $5 \sqrt{6}$ C. $10 \sqrt{2}$ D. $20 \sqrt{2}$

To determine the correct answer, we need to simplify the expression for the side length of the triangle.

Simplifying the Expression

Let's start by analyzing the given expression. We can assume that the expression is in the form of $a\sqrt{b}$, where $a$ and $b$ are integers.

To simplify the expression, we need to find the prime factorization of the number inside the square root. In this case, the number is 50.

Prime Factorization of 50

The prime factorization of 50 is:

50 = 2 × 5 × 5

Simplifying the Expression

Now that we have the prime factorization of 50, we can simplify the expression:

$4 \sqrt{50} = 4 \sqrt{2 × 5 × 5}$

Using the property of square roots, we can rewrite the expression as:

$4 \sqrt{50} = 4 \sqrt{2} \sqrt{5} \sqrt{5}$

Since $\sqrt{5} \sqrt{5} = 5$, we can simplify the expression further:

$4 \sqrt{50} = 4 \sqrt{2} \times 5$

$4 \sqrt{50} = 20 \sqrt{2}$

Conclusion

Based on our analysis, the simplest form of the expression for the side length of the triangle is:

$20 \sqrt{2}$

Therefore, the correct answer is:

D. $20 \sqrt{2}$

Why is this the simplest form?

The simplest form of an expression is the one that has the smallest possible coefficient and the simplest possible radical. In this case, the coefficient is 20, which is the smallest possible coefficient, and the radical is $\sqrt{2}$, which is the simplest possible radical.

What are the implications of this result?

The result has several implications:

• It shows that the side length of the triangle can be expressed in terms of the square root of 2.
• It provides a simplified expression for the side length, which can be used to perform calculations accurately.
• It demonstrates the importance of simplifying radical expressions in geometry.

Real-World Applications

The result has several real-world applications:

• In architecture, the side length of a triangle can be used to calculate the area of a building or a bridge.
• In engineering, the side length of a triangle can be used to calculate the stress and strain on a structure.
• In physics, the side length of a triangle can be used to calculate the distance and velocity of an object.

Conclusion

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Q: What is the significance of simplifying radical expressions in geometry?

A: Simplifying radical expressions in geometry is essential to understand the properties of triangles and to perform calculations accurately. It helps to identify the simplest form of an expression, which can be used to calculate the area, perimeter, and other properties of a triangle.

Q: How do you simplify a radical expression?

A: To simplify a radical expression, you need to find the prime factorization of the number inside the square root. Then, you can use the property of square roots to rewrite the expression and simplify it further.

Q: What is the difference between a simplified radical expression and a complex radical expression?

A: A simplified radical expression is one that has the smallest possible coefficient and the simplest possible radical. A complex radical expression, on the other hand, is one that has a large coefficient and a complex radical.

Q: Can you provide an example of a simplified radical expression?

A: Yes, an example of a simplified radical expression is $20 \sqrt{2}$. This expression is simplified because it has the smallest possible coefficient (20) and the simplest possible radical ($\sqrt{2}$).

Q: How do you determine the simplest form of a radical expression?

A: To determine the simplest form of a radical expression, you need to find the prime factorization of the number inside the square root. Then, you can use the property of square roots to rewrite the expression and simplify it further.

Q: What are some real-world applications of simplifying radical expressions in geometry?

A: Some real-world applications of simplifying radical expressions in geometry include:

• Calculating the area and perimeter of a building or a bridge
• Calculating the stress and strain on a structure
• Calculating the distance and velocity of an object

Q: Can you provide an example of a real-world application of simplifying radical expressions in geometry?

A: Yes, an example of a real-world application of simplifying radical expressions in geometry is calculating the area of a building. If the building has a triangular shape, you can use the simplified radical expression to calculate the area accurately.

Q: How do you calculate the area of a triangle using a simplified radical expression?

A: To calculate the area of a triangle using a simplified radical expression, you need to use the formula:

Area = (base × height) / 2

If the base and height are expressed as simplified radical expressions, you can simplify the expression further to calculate the area accurately.

Q: What are some common mistakes to avoid when simplifying radical expressions in geometry?

A: Some common mistakes to avoid when simplifying radical expressions in geometry include:

• Not finding the prime factorization of the number inside the square root
• Not using the property of square roots to rewrite the expression
• Not simplifying the expression further to get the simplest form

Q: How do you avoid these common mistakes when simplifying radical expressions in geometry?

A: To avoid these common mistakes when simplifying radical expressions in geometry, you need to:

• Find the prime factorization of the number inside the square root
• Use the property of square roots to rewrite the expression
• Simplify the expression further to get the simplest form

Conclusion

In conclusion, simplifying radical expressions in geometry is essential to understand the properties of triangles and to perform calculations accurately. By following the steps outlined in this article, you can simplify radical expressions and avoid common mistakes.