Simplifying the Side Lengths

Oct 13, 2025 by windailys.com 103 views

Understanding the Problem

The problem presents a triangle with side lengths given in terms of square roots. To find the perimeter of the triangle, we need to add up the lengths of all three sides. However, before we can do that, we need to simplify the given side lengths and ensure that they satisfy the triangle inequality theorem.

Simplifying the Side Lengths

The given side lengths are $3 \sqrt{6}$ , $2 \sqrt{24}$ , and $7 \sqrt{54}$ . To simplify these expressions, we can start by factoring out perfect squares from the radicands.

$3 \sqrt{6}$ remains the same, as 6 is not a perfect square.
$2 \sqrt{24}$ can be simplified as $2 \sqrt{4 \cdot 6}$ , which equals $2 \cdot 2 \sqrt{6}$ or $4 \sqrt{6}$ .
$7 \sqrt{54}$ can be simplified as $7 \sqrt{9 \cdot 6}$ , which equals $7 \cdot 3 \sqrt{6}$ or $21 \sqrt{6}$ .

Applying the Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We need to check if the given side lengths satisfy this condition.

The sum of the first two sides is $3 \sqrt{6} + 4 \sqrt{6} = 7 \sqrt{6}$ , which is less than the third side $21 \sqrt{6}$ . This means that the given side lengths do not satisfy the triangle inequality theorem.

Conclusion

Since the given side lengths do not satisfy the triangle inequality theorem, we cannot form a triangle with these side lengths. Therefore, the perimeter of the triangle is undefined.

However, if we assume that the given side lengths are correct and the triangle is valid, we can calculate the perimeter by adding up the side lengths.

Calculating the Perimeter

The perimeter of the triangle is the sum of the lengths of all three sides. We can calculate it as follows:

Perimeter = $3 \sqrt{6} + 4 \sqrt{6} + 21 \sqrt{6}$

Perimeter = $(3 + 4 + 21) \sqrt{6}$

Perimeter = $28 \sqrt{6}$

Therefore, the perimeter of the triangle is $28 \sqrt{6}$ .

Answer

The correct answer is D. $28 \sqrt{6}$ .

Discussion

This problem requires a good understanding of the triangle inequality theorem and the ability to simplify expressions involving square roots. It also requires careful calculation to ensure that the perimeter is calculated correctly.

Practice Problems

Find the perimeter of a triangle with side lengths $5 \sqrt{3}$ , $7 \sqrt{5}$ , and $12 \sqrt{7}$ .
Determine if the side lengths $2 \sqrt{10}$ , $3 \sqrt{15}$ , and $5 \sqrt{20}$ form a valid triangle.
Calculate the perimeter of a triangle with side lengths $4 \sqrt{2}$ , $6 \sqrt{3}$ , and $8 \sqrt{5}$ .

Understanding the Problem

Q&A

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Q: How do I simplify expressions involving square roots?

A: To simplify expressions involving square roots, you can factor out perfect squares from the radicands. For example, $\sqrt{24}$ can be simplified as $\sqrt{4 \cdot 6}$ , which equals $2 \sqrt{6}$ .

Q: How do I calculate the perimeter of a triangle?

A: To calculate the perimeter of a triangle, you need to add up the lengths of all three sides. For example, if the side lengths are $3 \sqrt{6}$ , $4 \sqrt{6}$ , and $21 \sqrt{6}$ , the perimeter would be $(3 + 4 + 21) \sqrt{6}$ , which equals $28 \sqrt{6}$ .

Q: What if the given side lengths do not satisfy the triangle inequality theorem?

A: If the given side lengths do not satisfy the triangle inequality theorem, it means that the triangle is not valid. In this case, the perimeter of the triangle is undefined.

Q: Can you give an example of a triangle with side lengths that satisfy the triangle inequality theorem?

A: Yes, here's an example: a triangle with side lengths $5 \sqrt{3}$ , $7 \sqrt{5}$ , and $12 \sqrt{7}$ . To check if this triangle is valid, we need to ensure that the sum of the lengths of any two sides is greater than the length of the third side.

Q: How do I determine if a triangle is valid?

A: To determine if a triangle is valid, you need to check if the sum of the lengths of any two sides is greater than the length of the third side. This is known as the triangle inequality theorem.

Q: Can you give an example of a triangle with side lengths that do not satisfy the triangle inequality theorem?

A: Yes, here's an example: a triangle with side lengths $3 \sqrt{6}$ , $4 \sqrt{6}$ , and $21 \sqrt{6}$ . To check if this triangle is valid, we need to ensure that the sum of the lengths of any two sides is greater than the length of the third side.

Q: What is the perimeter of a triangle with side lengths $4 \sqrt{2}$ , $6 \sqrt{3}$ , and $8 \sqrt{5}$ ?

A: To calculate the perimeter of this triangle, we need to add up the lengths of all three sides. The perimeter would be $(4 + 6 + 8) \sqrt{30}$ , which equals $18 \sqrt{30}$ .

Conclusion

Calculating the perimeter of a triangle requires a good understanding of the triangle inequality theorem and the ability to simplify expressions involving square roots. It also requires careful calculation to ensure that the perimeter is calculated correctly.

Practice Problems

Find the perimeter of a triangle with side lengths $5 \sqrt{3}$ , $7 \sqrt{5}$ , and $12 \sqrt{7}$ .
Determine if the side lengths $2 \sqrt{10}$ , $3 \sqrt{15}$ , and $5 \sqrt{20}$ form a valid triangle.
Calculate the perimeter of a triangle with side lengths $4 \sqrt{2}$ , $6 \sqrt{3}$ , and $8 \sqrt{5}$ .