Question

A triangle has sides 12.5 inches and 22 inches. What side length below would create a triangle with these two sides?(1 point)
Responses

4 inches
4 inches

9 inches
9 inches

11 inches
11 inches

35 inches

Answer 1

To determine which side length would allow the creation of a triangle with the two given sides (12.5 inches and 22 inches), we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Let’s denote the unknown side length by \( x \), with the other two sides being 12.5 inches and 22 inches. We can apply the triangle inequality conditions as follows:

1. \( 12.5 + 22 > x \)
- This simplifies to \( 34.5 > x \) or \( x < 34.5 \)

2. \( 12.5 + x > 22 \)
- This simplifies to \( x > 9.5 \)

3. \( 22 + x > 12.5 \)
- This simplifies to \( x > -9.5 \) (which is always true for positive \( x \))

Now we combine the results from the first two conditions:
- From \( x < 34.5 \)
- From \( x > 9.5 \)

Thus, \( x \) must satisfy:
\[ 9.5 < x < 34.5 \]

Now, let’s evaluate the provided options:

- **4 inches**: Not valid (since 4 is less than 9.5)
- **9 inches**: Not valid (since 9 is equal to 9.5)
- **11 inches**: Valid (since 11 is between 9.5 and 34.5)
- **35 inches**: Not valid (since 35 is greater than 34.5)

The only side length from the options given that would create a triangle with the other two sides is **11 inches**.