Which Classification Best Represents A Triangle With Side Lengths 10 In., 12 In., And 15 In.?A. Acute, Because $10^2 + 12^2 \ \textgreater \ 15^2$B. Acute, Because $12^2 + 15^2 \ \textgreater \ 10^2$C. Obtuse, Because
Introduction
In geometry, triangles are fundamental shapes that have been studied for centuries. One of the key aspects of triangle classification is determining whether a triangle is acute, right, or obtuse. In this article, we will explore the concept of triangle classification and determine which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.
What is Triangle Classification?
Triangle classification is the process of determining the type of triangle based on its angles and side lengths. There are three main types of triangles: acute, right, and obtuse.
- Acute Triangle: An acute triangle is a triangle with all angles less than 90 degrees. In other words, all the sides of an acute triangle are shorter than the hypotenuse (the side opposite the right angle).
- Right Triangle: A right triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle.
- Obtuse Triangle: An obtuse triangle is a triangle with one angle greater than 90 degrees. The side opposite the obtuse angle is called the hypotenuse, and it is always the longest side of the triangle.
How to Classify a Triangle
To classify a triangle, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
We can use this theorem to determine whether a triangle is acute, right, or obtuse.
Using the Pythagorean Theorem
Let's apply the Pythagorean theorem to the given triangle with side lengths 10 in., 12 in., and 15 in.
We can start by checking if the square of the longest side (15 in.) is equal to the sum of the squares of the other two sides (10 in. and 12 in.).
15^2 = 225 10^2 + 12^2 = 100 + 144 = 244
Since 225 is not equal to 244, we can conclude that the triangle is not a right triangle.
Next, we need to check if the square of the longest side (15 in.) is greater than the sum of the squares of the other two sides (10 in. and 12 in.).
15^2 = 225 10^2 + 12^2 = 100 + 144 = 244
Since 225 is greater than 244, we can conclude that the triangle is an obtuse triangle.
Conclusion
In conclusion, the triangle with side lengths 10 in., 12 in., and 15 in. is an obtuse triangle. This is because the square of the longest side (15 in.) is greater than the sum of the squares of the other two sides (10 in. and 12 in.).
References
- Pythagorean Theorem: The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
- Triangle Classification: Triangle classification is the process of determining the type of triangle based on its angles and side lengths. There are three main types of triangles: acute, right, and obtuse.
Frequently Asked Questions
- What is the Pythagorean theorem?
The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
- How do I classify a triangle?
To classify a triangle, you need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
You can then use this theorem to determine whether a triangle is acute, right, or obtuse.
Glossary
- Acute Triangle: An acute triangle is a triangle with all angles less than 90 degrees. In other words, all the sides of an acute triangle are shorter than the hypotenuse (the side opposite the right angle).
- Right Triangle: A right triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle.
- Obtuse Triangle: An obtuse triangle is a triangle with one angle greater than 90 degrees. The side opposite the obtuse angle is called the hypotenuse, and it is always the longest side of the triangle.
Further Reading
- Geometry: Geometry is the branch of mathematics that deals with the study of shapes, sizes, and positions of objects. It is a fundamental subject that has numerous applications in various fields, including architecture, engineering, and art.
- Trigonometry: Trigonometry is the branch of mathematics that deals with the study of triangles and their properties. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation.
Frequently Asked Questions: Triangle Classification =====================================================
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
Q: How do I classify a triangle?
A: To classify a triangle, you need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this can be expressed as:
a^2 + b^2 = c^2
You can then use this theorem to determine whether a triangle is acute, right, or obtuse.
Q: What is the difference between an acute triangle and an obtuse triangle?
A: An acute triangle is a triangle with all angles less than 90 degrees. In other words, all the sides of an acute triangle are shorter than the hypotenuse (the side opposite the right angle). An obtuse triangle, on the other hand, is a triangle with one angle greater than 90 degrees. The side opposite the obtuse angle is called the hypotenuse, and it is always the longest side of the triangle.
Q: How do I determine if a triangle is acute, right, or obtuse?
A: To determine if a triangle is acute, right, or obtuse, you need to use the Pythagorean theorem. If the square of the longest side (c) is equal to the sum of the squares of the other two sides (a and b), then the triangle is a right triangle. If the square of the longest side (c) is greater than the sum of the squares of the other two sides (a and b), then the triangle is an obtuse triangle. If the square of the longest side (c) is less than the sum of the squares of the other two sides (a and b), then the triangle is an acute triangle.
Q: What is the formula for calculating the area of a triangle?
A: The formula for calculating the area of a triangle is:
Area = (base × height) / 2
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. Mathematically, this can be expressed as:
Perimeter = a + b + c
Q: What is the difference between a scalene triangle and an isosceles triangle?
A: A scalene triangle is a triangle with all sides of different lengths. An isosceles triangle, on the other hand, is a triangle with two sides of equal length.
Q: How do I determine if a triangle is equilateral?
A: To determine if a triangle is equilateral, you need to check if all three sides are of equal length. If all three sides are of equal length, then the triangle is equilateral.
Q: What is the formula for calculating the height of a triangle?
A: The formula for calculating the height of a triangle is:
Height = (2 × Area) / base
Q: How do I calculate the volume of a triangular prism?
A: To calculate the volume of a triangular prism, you need to multiply the area of the base by the height. Mathematically, this can be expressed as:
Volume = (base × height) × length
Q: What is the difference between a right triangle and an obtuse triangle?
A: A right triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. An obtuse triangle, on the other hand, is a triangle with one angle greater than 90 degrees. The side opposite the obtuse angle is called the hypotenuse, and it is always the longest side of the triangle.
Q: How do I determine if a triangle is a 30-60-90 triangle?
A: To determine if a triangle is a 30-60-90 triangle, you need to check if the angles are in the ratio 1:2:√3. If the angles are in the ratio 1:2:√3, then the triangle is a 30-60-90 triangle.
Q: What is the formula for calculating the area of a right triangle?
A: The formula for calculating the area of a right triangle is:
Area = (base × height) / 2
Q: How do I calculate the perimeter of a right triangle?
A: To calculate the perimeter of a right triangle, you need to add up the lengths of all three sides. Mathematically, this can be expressed as:
Perimeter = a + b + c
Q: What is the difference between a scalene triangle and an isosceles triangle?
A: A scalene triangle is a triangle with all sides of different lengths. An isosceles triangle, on the other hand, is a triangle with two sides of equal length.
Q: How do I determine if a triangle is equilateral?
A: To determine if a triangle is equilateral, you need to check if all three sides are of equal length. If all three sides are of equal length, then the triangle is equilateral.
Q: What is the formula for calculating the height of a triangle?
A: The formula for calculating the height of a triangle is:
Height = (2 × Area) / base
Q: How do I calculate the volume of a triangular prism?
A: To calculate the volume of a triangular prism, you need to multiply the area of the base by the height. Mathematically, this can be expressed as:
Volume = (base × height) × length
Q: What is the difference between a right triangle and an obtuse triangle?
A: A right triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. An obtuse triangle, on the other hand, is a triangle with one angle greater than 90 degrees. The side opposite the obtuse angle is called the hypotenuse, and it is always the longest side of the triangle.
Q: How do I determine if a triangle is a 30-60-90 triangle?
A: To determine if a triangle is a 30-60-90 triangle, you need to check if the angles are in the ratio 1:2:√3. If the angles are in the ratio 1:2:√3, then the triangle is a 30-60-90 triangle.