Apply the triangle inequality theorem and determine which group of side measures will form a triangle.
answers:
8, 1, 7
12, 10, 22
9, 4, 3
2, 7, 6
To determine which group of side measures will form a triangle, we need to apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's evaluate each group of side measures:
Group 1: 8, 1, 7
The sum of the lengths of the first two sides is 8 + 1 = 9, which is greater than the length of the third side (7). Additionally, the sum of the lengths of the second and third sides is 1 + 7 = 8, which is greater than the length of the first side (8). Lastly, the sum of the lengths of the first and third sides is 8 + 7 = 15, which is greater than the length of the second side (1). Therefore, the side measures 8, 1, and 7 form a triangle.
Group 2: 12, 10, 22
The sum of the lengths of the first two sides is 12 + 10 = 22, which is equal to the length of the third side (22). According to the triangle inequality theorem, the sum of the lengths of any two sides must be greater than the length of the third side. Since the sum is equal, this group of side measures (12, 10, 22) does not form a triangle.
Group 3: 9, 4, 3
The sum of the lengths of the first two sides is 9 + 4 = 13, which is greater than the length of the third side (3). Additionally, the sum of the lengths of the second and third sides is 4 + 3 = 7, which is greater than the length of the first side (9). Lastly, the sum of the lengths of the first and third sides is 9 + 3 = 12, which is greater than the length of the second side (4). Therefore, the side measures 9, 4, and 3 form a triangle.
Group 4: 2, 7, 6
The sum of the lengths of the first two sides is 2 + 7 = 9, which is greater than the length of the third side (6). Additionally, the sum of the lengths of the second and third sides is 7 + 6 = 13, which is greater than the length of the first side (2). Lastly, the sum of the lengths of the first and third sides is 2 + 6 = 8, which is greater than the length of the second side (7). Therefore, the side measures 2, 7, and 6 form a triangle.
In summary, the groups of side measures that form a triangle are:
1. 8, 1, 7
2. 9, 4, 3
3. 2, 7, 6