Question

A triangle has one side length of 9
centimeters (cm) and another side length of 12
cm.

Which answers are possible lengths for the third side?

Select three that apply.

Responses

20
cm
20 cm

17
cm
17 cm

21
cm
21 cm

22
cm
22 cm

9
cm
9 cm

3
cm

Answer 1

To determine the possible lengths of the third side of a triangle when given two sides measuring 9 cm and 12 cm, we can use the triangle inequality theorem. This theorem 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 denote the sides as follows:
- Side A = 9 cm
- Side B = 12 cm
- Side C = the length of the third side we want to find.

The triangle inequalities can be expressed as:

1. \( A + B > C \)
2. \( A + C > B \)
3. \( B + C > A \)

Substituting the known values:

1. \( 9 + 12 > C \) which simplifies to \( C < 21 \)
2. \( 9 + C > 12 \) which simplifies to \( C > 3 \)
3. \( 12 + C > 9 \) which simplifies to \( C > -3 \) (which is always true for positive C)

From the inequalities, we find:
- \( C < 21 \)
- \( C > 3 \)

Thus, the length of the third side must be in the range:
\[ 3 < C < 21 \]

Now, we check the provided options against this range:

- **20 cm**: Valid (since \( 3 < 20 < 21 \))
- **17 cm**: Valid (since \( 3 < 17 < 21 \))
- **21 cm**: Invalid (since \( 21 \) is not less than \( 21 \))
- **22 cm**: Invalid (since \( 22 > 21 \))
- **9 cm**: Valid (since \( 3 < 9 < 21 \))
- **3 cm**: Invalid (since \( C > 3 \))

So, the possible lengths for the third side are:
- 20 cm
- 17 cm
- 9 cm

Therefore, the answers that apply are **20 cm, 17 cm, and 9 cm**.