Two sides of a triangle measure 4 cm and 9 cm. Determine which of the following choices cannot be the measure of the third side.
A: 8 cm
B:10 cm
C: 5 cm
D: 6 cm
C: 5 cm cannot be the measure of the third side.
To see why, we can use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.
In this case, the two given sides are 4 cm and 9 cm. So we need to check if the sum of these sides is greater than each of the answer choices.
- For choice A, 8 cm, the sum of 4 cm and 8 cm is 12 cm, which is greater than the third side.
- For choice B, 10 cm, the sum of 4 cm and 10 cm is 14 cm, which is greater than the third side.
- For choice D, 6 cm, the sum of 4 cm and 6 cm is 10 cm, which is greater than the third side.
However, for choice C, 5 cm, the sum of 4 cm and 5 cm is only 9 cm, which is not greater than the third side. Therefore, C cannot be the measure of the third side.
what a lot of words! Just recall that
9-4 < x < 9+4
5 < x < 13
so C:5 is not possible
To determine which of the given choices cannot be the measure of the third side of the triangle, we can use the triangle inequality theorem.
According to the theorem, the sum of the lengths of any two sides of the triangle must be greater than the length of the third side.
Let's check each choice one by one:
A: 8 cm
4 + 8 = 12 > 9, so 8 cm can be the measure of the third side.
B: 10 cm
4 + 10 = 14 > 9, so 10 cm can be the measure of the third side.
C: 5 cm
4 + 5 = 9, which is equal to the length of the third side. 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 9 is not greater than 9, 5 cm cannot be the measure of the third side.
D: 6 cm
4 + 6 = 10 > 9, so 6 cm can be the measure of the third side.
Therefore, the measure of the third side cannot be C: 5 cm.
To solve this problem, we can use 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 remaining side.
Let's check if the given side lengths (4 cm and 9 cm) satisfy the triangle inequality for each choice:
A: 4 + 9 > 8
13 > 8
Yes, 8 cm can be the length of the third side.
B: 4 + 9 > 10
13 > 10
Yes, 10 cm can be the length of the third side.
C: 4 + 9 > 5
13 > 5
Yes, 5 cm can be the length of the third side.
D: 4 + 9 > 6
13 > 6
Yes, 6 cm can be the length of the third side.
Therefore, all the given choices (A, B, C, and D) can be the measure of the third side of the triangle.