The Side Lengths Of A Triangle Are 9, 12, And 15. Is This A Right Triangle?A. Yes, Because 9 2 + 12 2 = 15 2 9^2 + 12^2 = 15^2 9 2 + 1 2 2 = 1 5 2 .B. Yes, Because 9 2 + 12 2 \textgreater 15 2 9^2 + 12^2 \ \textgreater \ 15^2 9 2 + 1 2 2 \textgreater 1 5 2 .C. No, Because 9 2 + 12 2 ≠ 15 2 9^2 + 12^2 \neq 15^2 9 2 + 1 2 2 = 1 5 2 .D. No, Because $9 +
Introduction
In geometry, a right triangle is a triangle with one angle that is a right angle, which is equal to 90 degrees. The side lengths of a right triangle follow the Pythagorean theorem, which states that 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 the side lengths of a triangle with sides 9, 12, and 15 form a right triangle.
Understanding the Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the side lengths of a right triangle. The theorem states that for any right triangle with sides of length a, b, and c, where c is the length of the hypotenuse, the following equation holds:
a^2 + b^2 = c^2
This equation can be used to determine whether a triangle is a right triangle or not. If the equation holds true, then the triangle is a right triangle.
Analyzing the Given Triangle
The given triangle has side lengths of 9, 12, and 15. To determine whether this triangle is a right triangle, we need to check if the Pythagorean theorem holds true for these side lengths. We can do this by plugging the values into the equation:
9^2 + 12^2 = 15^2
Calculating the Squares
To calculate the squares, we need to raise each side length to the power of 2:
9^2 = 81 12^2 = 144 15^2 = 225
Evaluating the Equation
Now that we have calculated the squares, we can evaluate the equation:
81 + 144 = 225
Conclusion
Based on the calculation, we can see that the equation holds true. The sum of the squares of the two shorter sides (81 + 144) is equal to the square of the longest side (225). Therefore, the triangle with side lengths 9, 12, and 15 is a right triangle.
Answer
The correct answer is A. Yes, because 9^2 + 12^2 = 15^2.
Why the Other Options are Incorrect
Option B is incorrect because it states that the sum of the squares of the two shorter sides is greater than the square of the longest side, which is not true in this case.
Option C is incorrect because it states that the sum of the squares of the two shorter sides is not equal to the square of the longest side, which is not true in this case.
Option D is incorrect because it states that the sum of the two shorter sides is not equal to the longest side, which is not relevant to the Pythagorean theorem.
Real-World Applications
The Pythagorean theorem has many real-world applications, including:
- Building design: Architects use the Pythagorean theorem to calculate the height of buildings and the length of shadows.
- Surveying: Surveyors use the Pythagorean theorem to calculate distances and angles between landmarks.
- Physics: Physicists use the Pythagorean theorem to calculate the trajectory of projectiles and the distance traveled by objects.
Conclusion
Introduction
In our previous article, we explored whether the side lengths of a triangle with sides 9, 12, and 15 form a right triangle. We concluded that the triangle is a right triangle because the Pythagorean theorem holds true for these side 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 fundamental concept in geometry that describes the relationship between the side lengths of a right triangle. The theorem states that for any right triangle with sides of length a, b, and c, where c is the length of the hypotenuse, the following equation holds:
a^2 + b^2 = c^2
Q: What is a right triangle?
A: A right triangle is a triangle with one angle that is a right angle, which is equal to 90 degrees. The side lengths of a right triangle follow the Pythagorean theorem.
Q: How do I determine if a triangle is a right triangle?
A: To determine if a triangle is a right triangle, you can use the Pythagorean theorem. If the equation a^2 + b^2 = c^2 holds true, then the triangle is a right triangle.
Q: What are the side lengths of a right triangle?
A: The side lengths of a right triangle are a, b, and c, where c is the length of the hypotenuse. The Pythagorean theorem states that a^2 + b^2 = c^2.
Q: Can a triangle have more than one right angle?
A: No, a triangle can only have one right angle. If a triangle has more than one right angle, it is not a triangle.
Q: Can a right triangle have all sides equal?
A: No, a right triangle cannot have all sides equal. If a triangle has all sides equal, it is an equilateral triangle, not a right triangle.
Q: Can a right triangle have all sides of equal length?
A: No, a right triangle cannot have all sides of equal length. If a triangle has all sides of equal length, it is an equilateral triangle, not a right triangle.
Q: What is the difference between a right triangle and an isosceles triangle?
A: A right triangle is a triangle with one right angle, while an isosceles triangle is a triangle with two sides of equal length. A right triangle can be isosceles, but not all isosceles triangles are right triangles.
Q: Can a right triangle be isosceles?
A: Yes, a right triangle can be isosceles. If a right triangle has two sides of equal length, it is an isosceles right triangle.
Q: Can a right triangle be equilateral?
A: No, a right triangle cannot be equilateral. If a triangle is equilateral, it is not a right triangle.
Q: Can a right triangle have a hypotenuse of length 0?
A: No, a right triangle cannot have a hypotenuse of length 0. The length of the hypotenuse must be greater than 0.
Q: Can a right triangle have a hypotenuse of length infinity?
A: No, a right triangle cannot have a hypotenuse of length infinity. The length of the hypotenuse must be a finite number.
Conclusion
In conclusion, the Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the side lengths of a right triangle. We have answered some frequently asked questions related to the Pythagorean theorem and right triangles. We hope this article has been helpful in understanding the concept of right triangles and the Pythagorean theorem.
Real-World Applications
The Pythagorean theorem has many real-world applications, including:
- Building design: Architects use the Pythagorean theorem to calculate the height of buildings and the length of shadows.
- Surveying: Surveyors use the Pythagorean theorem to calculate distances and angles between landmarks.
- Physics: Physicists use the Pythagorean theorem to calculate the trajectory of projectiles and the distance traveled by objects.
Practice Problems
- Determine if the triangle with side lengths 3, 4, and 5 is a right triangle.
- Determine if the triangle with side lengths 6, 8, and 10 is a right triangle.
- Determine if the triangle with side lengths 9, 12, and 15 is a right triangle.
- Calculate the length of the hypotenuse of a right triangle with side lengths 5 and 12.
- Calculate the length of the hypotenuse of a right triangle with side lengths 8 and 15.
Answer Key
- Yes, the triangle is a right triangle.
- Yes, the triangle is a right triangle.
- Yes, the triangle is a right triangle.
- The length of the hypotenuse is 13.
- The length of the hypotenuse is 17.