Question

Two sides of a triangle measure 6 1/2 cm and 10 mm. Apply the Triangle Inequality Theorem to find a possible measure of the third side.(1 point)
Responses

6 cm
6 cm

16 cm
16 cm

18 cm
18 cm

8 cm

Answer 1

To apply 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 first convert the measurements to the same unit.

The first side is given as 6 1/2 cm, which can be converted to an improper fraction:
\[
6 \frac{1}{2} \text{ cm} = \frac{13}{2} \text{ cm} \approx 6.5 \text{ cm}
\]
The second side is 10 mm, which we can convert to centimeters (since there are 10 mm in 1 cm):
\[
10 \text{ mm} = 1 \text{ cm}
\]

Now, we have the two sides of the triangle:
- Side 1 = 6.5 cm
- Side 2 = 1 cm

Let \( x \) be the length of the third side. According to the Triangle Inequality Theorem, we have the following inequalities:

1. \( 6.5 + 1 > x \)
2. \( 6.5 + x > 1 \)
3. \( 1 + x > 6.5 \)

Let's solve these inequalities one by one:

1. \( 6.5 + 1 > x \)
\[ 7.5 > x \]
(or \( x < 7.5 \))

2. \( 6.5 + x > 1 \)
\[ x > 1 - 6.5 \]
\[ x > -5.5 \]
(This condition will always be true since the side length must be positive)

3. \( 1 + x > 6.5 \)
\[ x > 6.5 - 1 \]
\[ x > 5.5 \]

Putting it all together, \( x \) must satisfy:
\[
5.5 < x < 7.5
\]

Now, reviewing the possible responses:
- 6 cm (fits since \( 5.5 < 6 < 7.5 \))
- 16 cm (does not fit)
- 18 cm (does not fit)
- 8 cm (does not fit)

So, the possible measure of the third side is **6 cm**.