Question
How to classify a triangle with side lengths 5, 7 and 10
Answers
GPT-4o mini
To classify a triangle based on its side lengths (5, 7, and 10), you can determine the type of triangle by checking the following:
1. **Determine whether it is a valid triangle**: The sum of the lengths of any two sides must be greater than the length of the third side. We can check this using the following inequalities:
- \( 5 + 7 > 10 \) (12 > 10, true)
- \( 5 + 10 > 7 \) (15 > 7, true)
- \( 7 + 10 > 5 \) (17 > 5, true)
Since all these conditions are satisfied, the side lengths form a valid triangle.
2. **Classify the triangle by the lengths of its sides**:
- If all three sides are of different lengths, it is a scalene triangle.
- If at least two sides are of the same length, it is an isosceles triangle.
- If all three sides are the same, it is an equilateral triangle.
In this case, since the side lengths (5, 7, and 10) are all different, it is classified as a **scalene triangle**.
3. **Classify by the angles**:
- You can determine if it is acute, right, or obtuse using the sides:
- Calculate \( a^2 + b^2 \) and compare it to \( c^2 \) (where \( c \) is the longest side):
- Here, let \( a = 5 \), \( b = 7 \), and \( c = 10 \).
- Calculate \( a^2 + b^2 = 5^2 + 7^2 = 25 + 49 = 74 \).
- Calculate \( c^2 = 10^2 = 100 \).
Now compare:
- If \( a^2 + b^2 < c^2 \), the triangle is obtuse.
- If \( a^2 + b^2 = c^2 \), the triangle is right.
- If \( a^2 + b^2 > c^2 \), the triangle is acute.
Since \( 74 < 100 \), the triangle is **obtuse**.
In summary, the triangle with side lengths 5, 7, and 10 is classified as a **scalene obtuse triangle**.
1. **Determine whether it is a valid triangle**: The sum of the lengths of any two sides must be greater than the length of the third side. We can check this using the following inequalities:
- \( 5 + 7 > 10 \) (12 > 10, true)
- \( 5 + 10 > 7 \) (15 > 7, true)
- \( 7 + 10 > 5 \) (17 > 5, true)
Since all these conditions are satisfied, the side lengths form a valid triangle.
2. **Classify the triangle by the lengths of its sides**:
- If all three sides are of different lengths, it is a scalene triangle.
- If at least two sides are of the same length, it is an isosceles triangle.
- If all three sides are the same, it is an equilateral triangle.
In this case, since the side lengths (5, 7, and 10) are all different, it is classified as a **scalene triangle**.
3. **Classify by the angles**:
- You can determine if it is acute, right, or obtuse using the sides:
- Calculate \( a^2 + b^2 \) and compare it to \( c^2 \) (where \( c \) is the longest side):
- Here, let \( a = 5 \), \( b = 7 \), and \( c = 10 \).
- Calculate \( a^2 + b^2 = 5^2 + 7^2 = 25 + 49 = 74 \).
- Calculate \( c^2 = 10^2 = 100 \).
Now compare:
- If \( a^2 + b^2 < c^2 \), the triangle is obtuse.
- If \( a^2 + b^2 = c^2 \), the triangle is right.
- If \( a^2 + b^2 > c^2 \), the triangle is acute.
Since \( 74 < 100 \), the triangle is **obtuse**.
In summary, the triangle with side lengths 5, 7, and 10 is classified as a **scalene obtuse triangle**.