Possible Measures for the Third Side of a Triangle Given Two Sides

To determine the possible length of the third side of a triangle given two sides, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let’s denote the two sides as ( a = 9 ) cm and ( b = 15 ) cm. We’ll denote the third side as ( c ). The inequalities we need to satisfy are:

1. ( a + b > c ) (9 + 15 > c)
2. ( a + c > b ) (9 + c > 15)
3. ( b + c > a ) (15 + c > 9)

Now, applying these to our options:

1. For option A (c = 23):

– ( 9 + 15 > 23 ) -> 24 > 23 (True)

– ( 9 + 23 > 15 ) -> 32 > 15 (True)

– ( 15 + 23 > 9 ) -> 38 > 9 (True)

2. For option B (c = 24):

– ( 9 + 15 > 24 ) -> 24 > 24 (False)

3. For option C (c = 25):

– ( 9 + 15 > 25 ) -> 24 > 25 (False)

4. For option