Question

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

Answer 1

To apply the triangle inequality theorem, we need to remember that for any triangle with sides \( a \), \( b \), and \( c \), the following must be true:

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

In this case, we have two sides measuring 1.5 cm and 10 mm. Let's convert everything to the same unit.

1.5 cm is already in centimeters, while 10 mm is equal to 1 cm (since 10 mm = 10/10 cm = 1 cm). Thus, we have:

- Side 1: \( a = 1.5 \) cm
- Side 2: \( b = 1 \) cm

Now, we can let the third side be \( c \).

Now applying the triangle inequality:

1. From \( a + b > c \):
\[
1.5 + 1 > c \implies 2.5 > c \implies c < 2.5
\]

2. From \( a + c > b \):
\[
1.5 + c > 1 \implies c > 1 - 1.5 \implies c > -0.5 \, \text{(always true since side lengths are positive)}
\]

3. From \( b + c > a \):
\[
1 + c > 1.5 \implies c > 1.5 - 1 \implies c > 0.5
\]

Combining the results, we have:

\[
0.5 < c < 2.5
\]

Now, we can evaluate each option given:

- 8 cm: not valid (greater than 2.5)
- 6 cm: not valid (greater than 2.5)
- 18 cm: not valid (greater than 2.5)
- 16 cm: not valid (greater than 2.5)

In conclusion, none of the given options (8 cm, 6 cm, 18 cm, 16 cm) can be a valid measure for the third side of the triangle, based on the triangle inequality theorem. The third side must be between 0.5 cm and 2.5 cm.