Which Set Of Numbers Can Represent The Side Lengths, In Inches, Of An Acute Triangle?A. 4, 5, 7 B. 5, 7, 8 C. 6, 7, 10 D. 7, 9, 12
In geometry, an acute triangle is a triangle in which all three angles are acute, meaning they are less than 90 degrees. The properties of acute triangles are crucial in understanding various mathematical concepts, including trigonometry and geometry. In this article, we will explore the properties of acute triangles and determine which set of numbers can represent the side lengths, in inches, of an acute triangle.
Properties of Acute Triangles
An acute triangle has three distinct properties:
- All angles are acute: As mentioned earlier, all three angles of an acute triangle are less than 90 degrees.
- All sides are of different lengths: In an acute triangle, all three sides are of different lengths.
- The sum of the squares of the two shorter sides is greater than the square of the longest side: This property is known as the Triangle Inequality Theorem.
Applying the Triangle Inequality Theorem
To determine which set of numbers can represent the side lengths of an acute triangle, we need to apply the Triangle Inequality Theorem. This theorem states that the sum of the squares of the two shorter sides must be greater than the square of the longest side.
Let's analyze each option:
Option A: 4, 5, 7
To determine if this set of numbers can represent the side lengths of an acute triangle, we need to apply the Triangle Inequality Theorem.
- 4^2 + 5^2 = 16 + 25 = 41
- 7^2 = 49
Since 41 is greater than 49, this set of numbers cannot represent the side lengths of an acute triangle.
Option B: 5, 7, 8
To determine if this set of numbers can represent the side lengths of an acute triangle, we need to apply the Triangle Inequality Theorem.
- 5^2 + 7^2 = 25 + 49 = 74
- 8^2 = 64
Since 74 is greater than 64, this set of numbers can represent the side lengths of an acute triangle.
Option C: 6, 7, 10
To determine if this set of numbers can represent the side lengths of an acute triangle, we need to apply the Triangle Inequality Theorem.
- 6^2 + 7^2 = 36 + 49 = 85
- 10^2 = 100
Since 85 is not greater than 100, this set of numbers cannot represent the side lengths of an acute triangle.
Option D: 7, 9, 12
To determine if this set of numbers can represent the side lengths of an acute triangle, we need to apply the Triangle Inequality Theorem.
- 7^2 + 9^2 = 49 + 81 = 130
- 12^2 = 144
Since 130 is not greater than 144, this set of numbers cannot represent the side lengths of an acute triangle.
Conclusion
In conclusion, only one set of numbers can represent the side lengths of an acute triangle. The correct answer is:
- Option B: 5, 7, 8
This set of numbers satisfies the Triangle Inequality Theorem, and therefore, it can represent the side lengths of an acute triangle.
Final Thoughts
In our previous article, we explored the properties of acute triangles and determined which set of numbers can represent the side lengths of an acute triangle. In this article, we will answer some frequently asked questions (FAQs) about acute triangles.
Q: What is an acute triangle?
A: An acute triangle is a triangle in which all three angles are acute, meaning they are less than 90 degrees.
Q: What are the properties of an acute triangle?
A: An acute triangle has three distinct properties:
- All angles are acute: All three angles of an acute triangle are less than 90 degrees.
- All sides are of different lengths: In an acute triangle, all three sides are of different lengths.
- The sum of the squares of the two shorter sides is greater than the square of the longest side: This property is known as the Triangle Inequality Theorem.
Q: How do I determine if a set of numbers can represent the side lengths of an acute triangle?
A: To determine if a set of numbers can represent the side lengths of an acute triangle, you need to apply the Triangle Inequality Theorem. This theorem states that the sum of the squares of the two shorter sides must be greater than the square of the longest side.
Q: What is the Triangle Inequality Theorem?
A: The Triangle Inequality Theorem states that the sum of the squares of the two shorter sides of a triangle must be greater than the square of the longest side.
Q: Can a right triangle be an acute triangle?
A: No, a right triangle cannot be an acute triangle. A right triangle has one angle that is 90 degrees, which means it is not an acute triangle.
Q: Can an obtuse triangle be an acute triangle?
A: No, an obtuse triangle cannot be an acute triangle. An obtuse triangle has one angle that is greater than 90 degrees, which means it is not an acute triangle.
Q: Can a scalene triangle be an acute triangle?
A: Yes, a scalene triangle can be an acute triangle. A scalene triangle is a triangle in which all three sides are of different lengths, and all three angles are acute.
Q: Can an isosceles triangle be an acute triangle?
A: Yes, an isosceles triangle can be an acute triangle. An isosceles triangle is a triangle in which two sides are of equal length, and all three angles are acute.
Q: Can a triangle with two sides of equal length and one side of a different length be an acute triangle?
A: Yes, a triangle with two sides of equal length and one side of a different length can be an acute triangle. This type of triangle is known as an isosceles triangle.
Q: Can a triangle with three sides of equal length be an acute triangle?
A: No, a triangle with three sides of equal length cannot be an acute triangle. This type of triangle is known as an equilateral triangle, and it is not an acute triangle.
Conclusion
In conclusion, acute triangles are an important concept in mathematics. By understanding the properties of acute triangles, we can determine which set of numbers can represent the side lengths of an acute triangle. We also answered some frequently asked questions (FAQs) about acute triangles, including questions about the Triangle Inequality Theorem, right triangles, obtuse triangles, scalene triangles, isosceles triangles, and equilateral triangles.