Question

two sides of a triangle measure 6 1/2 and 10 mm. apply the triangle inequality theorem to find a possible measure of the third side

Answer 1

The triangle inequality theorem states that for any triangle with sides \( a \), \( b \), and \( c \), the following inequalities must hold:

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

In this case, you have two sides of the triangle: \( a = 6.5 \) mm and \( b = 10 \) mm. Let's denote the third side as \( c \).

Now, we can apply the triangle inequality:

1. \( a + b > c \)
\[
6.5 + 10 > c \implies 16.5 > c \implies c < 16.5
\]

2. \( a + c > b \)
\[
6.5 + c > 10 \implies c > 10 - 6.5 \implies c > 3.5
\]

3. \( b + c > a \)
\[
10 + c > 6.5 \implies c > 6.5 - 10 \implies c > -3.5
\]
(This inequality is always true and does not provide additional constraints.)

Combining the inequalities from the first two conditions, we find:
\[
3.5 < c < 16.5
\]

Thus, a possible measure for the third side \( c \) must be greater than 3.5 mm and less than 16.5 mm. For example, a possible measure of the third side could be \( 5 \) mm or \( 15 \) mm.