Two sides of a triangle measure 3.7 and 8.2 apply the triangle i n e q u a l i t y t h e o r e m to find a possible measure of the third side a 3.5 B 5.5 c 4.5 the 4.2
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's consider the possible combinations using the given sides:
1) 3.7 + 8.2 > 3rd side: The sum of 3.7 and 8.2 is 11.9, which is greater than any of the options given. Therefore, 11.9 is a valid measurement for the third side.
2) 3.7 + 3.5 > 3rd side: The sum of 3.7 and 3.5 is 7.2, which is less than 8.2. Therefore, 3.5 is not a valid measurement for the third side.
3) 8.2 + 3.5 > 3rd side: The sum of 8.2 and 3.5 is 11.7, which is greater than any of the options given. Therefore, 11.7 is a valid measurement for the third side.
4) 3.7 + 5.5 > 3rd side: The sum of 3.7 and 5.5 is 9.2, which is less than 8.2. Therefore, 5.5 is not a valid measurement for the third side.
5) 8.2 + 5.5 > 3rd side: The sum of 8.2 and 5.5 is 13.7, which is greater than any of the options given. Therefore, 13.7 is a valid measurement for the third side.
6) 3.7 + 4.5 > 3rd side: The sum of 3.7 and 4.5 is 8.2, which is equal to the second given side. Therefore, 4.5 is not a valid measurement for the third side.
7) 8.2 + 4.5 > 3rd side: The sum of 8.2 and 4.5 is 12.7, which is greater than any of the options given. Therefore, 12.7 is a valid measurement for the third side.
In summary, the possible measures for the third side are 11.9, 11.7, and 12.7. Thus, options A (3.5) and C (4.5) are not valid measurements for the third side. The valid measurement from the given options is 4.2 (option B).