Question

Find the range of values for the third side of the triangle x when two sides are 7 and the bottom side isn't shown

Answer 1

To determine the range of values for the third side \( x \) of a triangle when the other two sides are given as 7 each, we can apply the triangle inequality theorem. This 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, let the two known sides \( a \) and \( b \) both be 7, and let \( c \) be the unknown side \( x \). We can set up the inequalities as follows:

1. \( 7 + 7 > x \) → \( 14 > x \) → \( x < 14 \)
2. \( 7 + x > 7 \) → \( x > 0 \)
3. \( 7 + x > 7 \) → \( x > 0 \) (This inequality does not provide new information.)

Now, combining the inequalities from the triangle inequality theorem gives us:
- \( x < 14 \)
- \( x > 0 \)

Thus, the range of values for the third side \( x \) is:

\[
0 < x < 14
\]

Hence, \( x \) must be a positive number less than 14.