Question

Two sides of a triangle measured 6.5 cm and 10 mm apply the triangle inequality theorem to find a possible measure for the third side
8 cm
6 cm
18 cm
16 cm

Answer 1

To use the triangle inequality theorem, we need to ensure 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 convert all measurements to the same unit. We'll convert everything to centimeters:

- 6.5 cm (already in cm)
- 10 mm = 1 cm (since 10 mm = 1 cm)

Now we have two sides:
- Side A = 6.5 cm
- Side B = 1 cm

Let the length of the third side be \( C \).

According to the triangle inequality theorem:

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

Plugging in our known values:

1. \( 6.5 + 1 > C \)
- \( 7.5 > C \)
- So, \( C < 7.5 \)

2. \( 6.5 + C > 1 \)
- \( 6.5 + C > 1 \) is always true since \( C \) must be positive.

3. \( 1 + C > 6.5 \)
- \( C > 5.5 \)

From inequalities 1 and 3, we have:
- \( 5.5 < C < 7.5 \)

Now let's compare this range to the options given:
- 8 cm
- 6 cm
- 18 cm
- 16 cm

The only value that falls within the range of \( 5.5 < C < 7.5 \) is **6 cm**. Thus, the possible measure for the third side is **6 cm**.