Given the lengths of the sides of a triangle, identify the triangle as acute, right, or obtuse.
37,12,34
5,12,13
20,21,28
given the three sides, with c being the largest, you know that the triangle is a right triangle if
c^2 = a^2+b^2
if c^2 < a^2+b^2, then it's acute, and obtuse otherwise.
So, start checking.
Thank you Steve.
To determine whether a triangle is acute, right, or obtuse, we need to analyze the relationship between the squares of the side lengths.
1. Triangle with side lengths 37, 12, 34:
To determine the relationship, we need to check if the sum of the squares of the two smaller side lengths is greater than the square of the longest side.
a² + b² > c², where a, b, c are the side lengths.
Considering the side lengths:
37² + 12² = 1633
34² = 1156
Since 1633 is greater than 1156, we can conclude that the triangle with sides 37, 12, and 34 is an acute triangle.
2. Triangle with side lengths 5, 12, 13:
Applying the same condition:
a² + b² > c²
5² + 12² = 169
13² = 169
Since 169 is equal to 169, we can conclude that the triangle with sides 5, 12, and 13 is a right triangle.
3. Triangle with side lengths 20, 21, 28:
Again, applying the same condition:
a² + b² > c²
20² + 21² = 841 + 441 = 1282
28² = 784
Since 1282 is greater than 784, we can conclude that the triangle with sides 20, 21, and 28 is an obtuse triangle.
To summarize:
- Triangle with sides 37, 12, and 34 is an acute triangle.
- Triangle with sides 5, 12, and 13 is a right triangle.
- Triangle with sides 20, 21, and 28 is an obtuse triangle.
To determine whether a given triangle is acute, right, or obtuse, we need to check the relationship between the squares of the lengths of its sides. Here's how you can identify the type of triangle for each case:
1) Triangle with side lengths 37, 12, 34:
First, we need to find the squares of the side lengths. The squared lengths of the sides are:
37^2 = 1369
12^2 = 144
34^2 = 1156
Next, we need to check the relationship between these squared lengths. In an acute triangle, the sum of the squares of the smaller two sides would be greater than the square of the largest side. In a right triangle, the sum of the squares of the two smaller sides should be equal to the square of the largest side. However, in an obtuse triangle, the sum of the squares of the smaller two sides would be less than the square of the largest side.
For the given side lengths (1369, 144, 1156), the sum of the squares of the smaller two sides is 144 + 1156 = 1300, which is less than 1369. Therefore, the triangle with side lengths 37, 12, and 34 is an obtuse triangle.
2) Triangle with side lengths 5, 12, 13:
Similarly, we find the squared lengths of the sides:
5^2 = 25
12^2 = 144
13^2 = 169
Now we check the relationship between these squared lengths. The sum of the squares of the smaller two sides is 25 + 144 = 169, which is equal to the square of the longest side (13^2). Hence, the triangle with side lengths 5, 12, and 13 is a right triangle.
3) Triangle with side lengths 20, 21, 28:
Finding the squares of the side lengths:
20^2 = 400
21^2 = 441
28^2 = 784
Checking the relationship between these squared lengths, the sum of the squares of the smaller two sides is 400 + 441 = 841, which is greater than the square of the largest side (28^2). Therefore, the triangle with side lengths 20, 21, and 28 is an acute triangle.
In summary:
- Triangle with side lengths 37, 12, and 34 is an obtuse triangle.
- Triangle with side lengths 5, 12, and 13 is a right triangle.
- Triangle with side lengths 20, 21, and 28 is an acute triangle.