Can You Form A Right Triangle With The Three Lengths Given? Why Or Why Not?$4 , M, 6 , M, 7 , M$A. Yes; $4^2 + 6^2 \neq 7^2$B. Yes; $4 + 6 = 7$C. No; $4^2 + 6^2 \neq 7^2$D. No; $4 + 6 \neq 7$

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Introduction

When it comes to forming a right triangle with three given lengths, there are certain conditions that must be met. The Pythagorean theorem is a fundamental concept in geometry that helps us determine whether a triangle is a right triangle or not. In this article, we will explore the concept of the Pythagorean theorem and how it applies to the given lengths of 4 meters, 6 meters, and 7 meters.

Understanding the Pythagorean Theorem

The Pythagorean theorem is a mathematical concept that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed mathematically as:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Given Lengths

Now that we have a good understanding of the Pythagorean theorem, let's apply it to the given lengths of 4 meters, 6 meters, and 7 meters. We can use the theorem to determine whether these lengths can form a right triangle.

Option A: 4^2 + 6^2 ≠ 7^2

One of the options is to calculate the squares of the given lengths and compare them to see if they satisfy the Pythagorean theorem. Let's calculate the squares of the given lengths:

4^2 = 16 6^2 = 36 7^2 = 49

Now, let's compare the sum of the squares of the two shorter sides (4 meters and 6 meters) to the square of the longest side (7 meters):

16 + 36 = 52 52 ≠ 49

As we can see, the sum of the squares of the two shorter sides (52) is not equal to the square of the longest side (49). Therefore, option A is incorrect.

Option B: 4 + 6 = 7

Another option is to check if the sum of the two shorter sides is equal to the longest side. Let's calculate the sum of the two shorter sides:

4 + 6 = 10 10 ≠ 7

As we can see, the sum of the two shorter sides (10) is not equal to the longest side (7). Therefore, option B is incorrect.

Option C: 4^2 + 6^2 ≠ 7^2

We have already calculated the squares of the given lengths in the previous section. Let's recall the results:

4^2 = 16 6^2 = 36 7^2 = 49

Now, let's compare the sum of the squares of the two shorter sides (4 meters and 6 meters) to the square of the longest side (7 meters):

16 + 36 = 52 52 ≠ 49

As we can see, the sum of the squares of the two shorter sides (52) is not equal to the square of the longest side (49). Therefore, option C is correct.

Conclusion

In conclusion, we have applied the Pythagorean theorem to the given lengths of 4 meters, 6 meters, and 7 meters. We have calculated the squares of the given lengths and compared them to see if they satisfy the Pythagorean theorem. Based on our calculations, we have determined that option C is the correct answer. Therefore, it is not possible to form a right triangle with the three lengths given.

Frequently Asked Questions

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical concept that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I apply the Pythagorean theorem to a given set of lengths?

A: To apply the Pythagorean theorem, you need to calculate the squares of the given lengths and compare them to see if they satisfy the theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the lengths can form a right triangle.

Q: What are the conditions for a triangle to be a right triangle?

A: A triangle is a right triangle if the sum of the squares of the two shorter sides is equal to the square of the longest side.

References

Further Reading

Related Articles

  • [1] "Can You Form a Right Triangle with the Three Lengths Given? Why or Why Not?" (This article)
  • [2] "Understanding the Pythagorean Theorem"
  • [3] "Applying the Pythagorean Theorem to Real-World Problems"

Introduction

In our previous article, we explored the concept of the Pythagorean theorem and how it applies to the given lengths of 4 meters, 6 meters, and 7 meters. We determined that it is not possible to form a right triangle with these lengths. In this article, we will answer some frequently asked questions related to the Pythagorean theorem and right triangles.

Q&A

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical concept that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I apply the Pythagorean theorem to a given set of lengths?

A: To apply the Pythagorean theorem, you need to calculate the squares of the given lengths and compare them to see if they satisfy the theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the lengths can form a right triangle.

Q: What are the conditions for a triangle to be a right triangle?

A: A triangle is a right triangle if the sum of the squares of the two shorter sides is equal to the square of the longest side.

Q: Can I form a right triangle with any three lengths?

A: No, you cannot form a right triangle with any three lengths. The Pythagorean theorem only applies to right-angled triangles, and the lengths must satisfy the theorem in order to form a right triangle.

Q: What happens if the sum of the squares of the two shorter sides is greater than the square of the longest side?

A: If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the lengths cannot form a right triangle.

Q: What happens if the sum of the squares of the two shorter sides is less than the square of the longest side?

A: If the sum of the squares of the two shorter sides is less than the square of the longest side, then the lengths cannot form a right triangle.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse. If you know the lengths of the two shorter sides, you can calculate the square of the hypotenuse by subtracting the sum of the squares of the two shorter sides from the square of the longest side.

Q: Can I use the Pythagorean theorem to find the length of one of the shorter sides?

A: Yes, you can use the Pythagorean theorem to find the length of one of the shorter sides. If you know the lengths of the other two sides, you can calculate the length of the missing side by using the theorem.

Examples

Example 1: Finding the length of the hypotenuse

Suppose we have a right triangle with one side of length 3 meters and the other side of length 4 meters. We want to find the length of the hypotenuse. Using the Pythagorean theorem, we can calculate the square of the hypotenuse as follows:

3^2 + 4^2 = 9 + 16 = 25 √25 = 5

Therefore, the length of the hypotenuse is 5 meters.

Example 2: Finding the length of one of the shorter sides

Suppose we have a right triangle with one side of length 5 meters and the hypotenuse of length 10 meters. We want to find the length of the other side. Using the Pythagorean theorem, we can calculate the length of the missing side as follows:

5^2 + x^2 = 10^2 25 + x^2 = 100 x^2 = 75 x = √75

Therefore, the length of the missing side is √75 meters.

Conclusion

In conclusion, the Pythagorean theorem is a powerful tool for determining whether a triangle is a right triangle or not. By applying the theorem to a given set of lengths, we can determine whether the lengths can form a right triangle or not. We have answered some frequently asked questions related to the Pythagorean theorem and right triangles, and provided examples of how to use the theorem to find the length of the hypotenuse or one of the shorter sides.

Frequently Asked Questions

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical concept that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I apply the Pythagorean theorem to a given set of lengths?

A: To apply the Pythagorean theorem, you need to calculate the squares of the given lengths and compare them to see if they satisfy the theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the lengths can form a right triangle.

Q: What are the conditions for a triangle to be a right triangle?

A: A triangle is a right triangle if the sum of the squares of the two shorter sides is equal to the square of the longest side.

References

Further Reading

Related Articles

  • [1] "Can You Form a Right Triangle with the Three Lengths Given? Why or Why Not?" (This article)
  • [2] "Understanding the Pythagorean Theorem"
  • [3] "Applying the Pythagorean Theorem to Real-World Problems"