To classify the triangle based on its side lengths, we can use the triangle inequality theorem and properties of triangles.
In this case, the sides are 20 cm, 99 cm, and 108 cm. First, let’s check if these sides can form a triangle using the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let’s check:
- 20 cm + 99 cm = 119 cm > 108 cm (True)
- 20 cm + 108 cm = 128 cm > 99 cm (True)
- 99 cm + 108 cm = 207 cm > 20 cm (True)
Since all three conditions are satisfied, we can confirm that these lengths can indeed form a triangle.
Next, to classify the triangle, we will utilize the lengths of the sides:
1. An acute triangle has all angles less than 90 degrees.
2. A right triangle has one angle equal to 90 degrees.
3. An obtuse triangle has one angle greater than 90 degrees.
We can determine the type of triangle by using the sides’ lengths in relation to the largest side. Here, the largest side is 108 cm.
To see if the triangle is acute, right, or obtuse, we can check the following condition:
If the square of the longest side (c) is greater than the sum of the squares of the other two sides (a and b), the triangle is obtuse. If it is equal, the triangle is right, and if it is less, the triangle is acute.
Let’s calculate:
c² = 108² = 11664
a² + b² = 20² + 99² = 400 + 9801 = 10201
Now, we compare:
11664 > 10201
This inequality shows that the square of the longest side is greater than the sum of the squares of the other two sides.
Therefore, the triangle with sides measuring 20 cm, 99 cm, and 108 cm is classified as an obtuse triangle.