Which Classification Best Represents A Triangle With Side Lengths $10 \text{ In.}, 12 \text{ In.}, \text{and } 15 \text{ In.}$?A. Acute, Because $10^2 + 12^2 \ \textgreater \ 15^2$B. Acute, Because $12^2 + 15^2 \
Introduction
In geometry, triangles are classified based on their angles and side lengths. Understanding the properties of triangles is crucial in various mathematical and real-world applications. In this article, we will explore the classification of triangles and determine which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.
Types of Triangles
Triangles can be classified into three main categories: acute, right, and obtuse triangles. The classification of a triangle is determined by the measure of its angles and the relationship between its side lengths.
Acute Triangle
An acute triangle is a triangle with all angles measuring less than 90 degrees. In an acute triangle, 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.
Right Triangle
A right triangle is a triangle with one angle measuring exactly 90 degrees. In a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side. This property is also known as the Pythagorean Theorem.
Obtuse Triangle
An obtuse triangle is a triangle with one angle measuring greater than 90 degrees. In an obtuse triangle, the sum of the squares of the two shorter sides is less than the square of the longest side.
Determining the Classification of a Triangle
To determine the classification of a triangle, we need to examine the relationship between its side lengths. We can use the Triangle Inequality Theorem to determine if a triangle is acute or obtuse.
Using the Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the squares of the two shorter sides of a triangle is greater than the square of the longest side. Mathematically, this can be expressed as:
a^2 + b^2 > c^2
where a and b are the lengths of the two shorter sides, and c is the length of the longest side.
Applying the Triangle Inequality Theorem
Let's apply the Triangle Inequality Theorem to the given triangle with side lengths 10 in., 12 in., and 15 in.
We can write the following inequalities:
- 10^2 + 12^2 > 15^2
- 10^2 + 15^2 > 12^2
- 12^2 + 15^2 > 10^2
Evaluating the Inequalities
Now, let's evaluate the inequalities:
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10^2 + 12^2 = 100 + 144 = 244
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15^2 = 225
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244 > 225 (True)
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10^2 + 15^2 = 100 + 225 = 325
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12^2 = 144
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325 > 144 (True)
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12^2 + 15^2 = 144 + 225 = 369
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10^2 = 100
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369 > 100 (True)
Conclusion
Based on the evaluation of the inequalities, we can conclude that the triangle with side lengths 10 in., 12 in., and 15 in. is an acute triangle. The sum of the squares of the two shorter sides (10^2 + 12^2) is greater than the square of the longest side (15^2).
Final Answer
The final answer is: A. Acute, because 10^2 + 12^2 > 15^2
Introduction
In our previous article, we explored the classification of triangles and determined which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in. In this article, we will answer some frequently asked questions related to triangle classification.
Q: What is the difference between an acute and an obtuse triangle?
A: An acute triangle is a triangle with all angles measuring less than 90 degrees, while an obtuse triangle is a triangle with one angle measuring greater than 90 degrees.
Q: How do I determine if a triangle is acute or obtuse?
A: To determine if a triangle is acute or obtuse, you can use the Triangle Inequality Theorem, which states that the sum of the squares of the two shorter sides of a triangle is greater than the square of the longest side.
Q: What is the Pythagorean Theorem?
A: The Pythagorean Theorem is a mathematical formula that states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side. Mathematically, this can be expressed as:
a^2 + b^2 = c^2
where a and b are the lengths of the two shorter sides, and c is the length of the longest side.
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 sum of the squares of the two shorter sides is equal to the square of the longest side, then the triangle is a right triangle.
Q: What is the difference between a scalene and an isosceles triangle?
A: A scalene triangle is a triangle with all sides of different lengths, while an isosceles triangle is a triangle with two sides of equal length.
Q: How do I determine if a triangle is scalene or isosceles?
A: To determine if a triangle is scalene or isosceles, you can compare the lengths of the sides. If all sides are of different lengths, then the triangle is scalene. If two sides are of equal length, then the triangle is isosceles.
Q: What is the difference between an equilateral and an equiangular triangle?
A: An equilateral triangle is a triangle with all sides of equal length, while an equiangular triangle is a triangle with all angles measuring 60 degrees.
Q: How do I determine if a triangle is equilateral or equiangular?
A: To determine if a triangle is equilateral or equiangular, you can compare the lengths of the sides and the measures of the angles. If all sides are of equal length, then the triangle is equilateral. If all angles measure 60 degrees, then the triangle is equiangular.
Conclusion
In this article, we answered some frequently asked questions related to triangle classification. We hope that this article has provided you with a better understanding of the properties of triangles and how to classify them.
Final Answer
The final answer is: There is no final numerical answer to this article, as it is a Q&A article.