triangle

one side of the triangle is 7

Find the range of values for third side of the triangle, x, based on the diagram.
_ < x < _

1 answer

To find the range of values for the third side \( x \) of a triangle when one side is 7, we can use the triangle inequality theorem. The triangle inequality states 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 denote the sides of the triangle as follows:

  • Side 1 = 7
  • Side 2 = \( b \) (unknown)
  • Side 3 = \( x \) (the side we want to find)

According to the triangle inequality, we have three conditions to satisfy:

  1. \( 7 + b > x \)
  2. \( 7 + x > b \)
  3. \( b + x > 7 \)

Given that \( b \) is a positive value, we can analyze these inequalities.

From the inequalities:

  1. Rearranging \( 7 + b > x \) gives \( x < 7 + b \).
  2. Rearranging \( 7 + x > b \) gives \( b < 7 + x \).
  3. Rearranging \( b + x > 7 \) gives \( b > 7 - x \).

Since \( b \) must be positive, we can combine the inequalities as follows:

  • Let's assume \( b \) is at least \( 1 \) (the minimum length for a side):

From \( b > 7 - x \),

  • If \( b = 1 \), we find \( 1 > 7 - x\) or \( x > 6\).
  • If \( b \) can take larger values, this further constrains \( x \).

From \( x < 7 + b \), as \( b \) can be any positive value but must also relate with \( x\) being less than a certain upper limit.

Where the tightest constraint comes from the case \( b < 7 + x\):

Now, combining these together for ranges involving \( b\):

  • The conditions imply that for \( x \): \[ 6 < x < \infty \] As \( b\) can technically be indefinitely large, while \( x < b + 7\).

But keeping practical side constraints, let's assume a scenario where both sides remain positive.

This leads us effectively to conclude that:

  • The stricter triangle constraint is dependent on actual b, but with non-static manipulations \( x\) must be reasonably closed by practical upper limits.

In practical terms:

  • Assuming a maximum constraint for visible values of triangle, and where this meets angle constraints gives (for visible reading): Summary: Assuming just one static, brings us also beyond: \[ 6 < x < 14\] Could represent reasonable assumption bounds where interactions of sizes meet or allow triangular forms.

Thus, \( _ < x < _ \) implying inferred limits gives result: Final Representation suggesting: \[ 6 < x < 14 \]