Can You Form A Right Triangle With The Three Lengths Given: 7 In., 24 In., 25 In.?A. No; ${ 7 + 24 = 25\$} B. Yes; ${ 7 + 24 = 25\$} C. No; ${ 7^2 + 24^2 \neq 25^2\$} D. Yes; ${ 7^2 + 24^2 = 25^2\$}
Introduction
In geometry, a right triangle is a triangle in which one of the angles is a right angle (90 degrees). The Pythagorean theorem is a fundamental concept in geometry that helps us determine whether a triangle is a right triangle or not. The theorem 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. In this article, we will explore whether it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in.
Understanding the Pythagorean Theorem
The Pythagorean theorem is a mathematical concept that has been used for centuries to determine whether a triangle is a right triangle or not. The theorem is named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with its discovery. The theorem 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
To determine whether it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in., we can apply the Pythagorean theorem to the given lengths. Let's assume that the length of 7 in. is one of the sides that form the right angle, the length of 24 in. is the other side that forms the right angle, and the length of 25 in. is the hypotenuse.
Calculating the Squares of the Lengths
To apply the Pythagorean theorem, we need to calculate the squares of the lengths of the three sides. The square of the length of 7 in. is:
7^2 = 49
The square of the length of 24 in. is:
24^2 = 576
The square of the length of 25 in. is:
25^2 = 625
Checking if the Pythagorean Theorem is Satisfied
Now that we have calculated the squares of the lengths of the three sides, we can check if the Pythagorean theorem is satisfied. According to the theorem, the sum of the squares of the lengths of the two sides that form the right angle should be equal to the square of the length of the hypotenuse. Let's check if this is the case:
49 + 576 = 625
As we can see, the sum of the squares of the lengths of the two sides that form the right angle (49 and 576) is equal to the square of the length of the hypotenuse (625). This means that the Pythagorean theorem is satisfied, and it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in.
Conclusion
In conclusion, we have applied the Pythagorean theorem to the three lengths given: 7 in., 24 in., and 25 in. We have calculated the squares of the lengths of the three sides and checked if the Pythagorean theorem is satisfied. As we have seen, the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse, which means that it is possible to form a right triangle with the three lengths given. Therefore, the correct answer is:
D. Yes; ${7^2 + 24^2 = 25^2\$}
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 triangle?
A: To apply the Pythagorean theorem to a given triangle, you need to calculate the squares of the lengths of the three sides and check if the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse.
Q: What are the three lengths given in this problem?
A: The three lengths given in this problem are 7 in., 24 in., and 25 in.
Q: Is it possible to form a right triangle with the three lengths given?
A: Yes, it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in.
References
- "Pythagorean Theorem." Math Open Reference, mathopenref.com/pythagorean_theorem.html.
- "Right Triangle." Math Is Fun, mathisfun.com/algebra/right-triangle.html.
- "Pythagorean Theorem Formula." Formula List, formulalist.com/pythagorean-theorem-formula/.
Introduction
In our previous article, we explored whether it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in. We applied the Pythagorean theorem to the given lengths and found that it is indeed 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 triangle?
A: To apply the Pythagorean theorem to a given triangle, you need to calculate the squares of the lengths of the three sides and check if the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse.
Q: What are the three lengths given in this problem?
A: The three lengths given in this problem are 7 in., 24 in., and 25 in.
Q: Is it possible to form a right triangle with the three lengths given?
A: Yes, it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in.
Q: What is the formula for the Pythagorean theorem?
A: The formula for the Pythagorean theorem is:
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.
Q: How do I calculate the squares of the lengths of the sides?
A: To calculate the squares of the lengths of the sides, you need to multiply the length of each side by itself. For example, if the length of a side is 5 in., the square of the length is 5^2 = 25.
Q: What is the difference between a right triangle and an oblique triangle?
A: A right triangle is a triangle in which one of the angles is a right angle (90 degrees). An oblique triangle is a triangle in which none of the angles are right angles.
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 sides that form the right angle, you can use the formula to find the length of the hypotenuse.
Q: Can I use the Pythagorean theorem to find the length of one of the sides?
A: Yes, you can use the Pythagorean theorem to find the length of one of the sides. If you know the lengths of the other two sides, you can use the formula to find the length of the third side.
Conclusion
In conclusion, we have answered some frequently asked questions related to the Pythagorean theorem and right triangles. We have explained the formula for the Pythagorean theorem, how to apply it to a given triangle, and how to calculate the squares of the lengths of the sides. We have also discussed the difference between a right triangle and an oblique triangle, and how to use the Pythagorean theorem to find the length of the hypotenuse or one of the 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 triangle?
A: To apply the Pythagorean theorem to a given triangle, you need to calculate the squares of the lengths of the three sides and check if the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse.
Q: What are the three lengths given in this problem?
A: The three lengths given in this problem are 7 in., 24 in., and 25 in.
Q: Is it possible to form a right triangle with the three lengths given?
A: Yes, it is possible to form a right triangle with the three lengths given: 7 in., 24 in., and 25 in.
References
- "Pythagorean Theorem." Math Open Reference, mathopenref.com/pythagorean_theorem.html.
- "Right Triangle." Math Is Fun, mathisfun.com/algebra/right-triangle.html.
- "Pythagorean Theorem Formula." Formula List, formulalist.com/pythagorean-theorem-formula/.