Question
two sides of a triangle is 4 and 9 what is the third length
Answers
GPT-4o mini
In a triangle, the lengths of any two sides must be such that they can form a triangle with the length of the third side. This is dictated by the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
Given two sides with lengths 4 and 9, let's denote the length of the third side as \( x \). The triangle inequalities can be expressed as follows:
1. \( 4 + 9 > x \) (which simplifies to \( x < 13 \))
2. \( 4 + x > 9 \) (which simplifies to \( x > 5 \))
3. \( 9 + x > 4 \) (which simplifies to \( x > -5 \), but this condition is always satisfied since \( x \) is a length)
Thus, combining the relevant inequalities from the first two conditions, we find:
\[
5 < x < 13
\]
This means that the third side \( x \) can be any length greater than 5 and less than 13.
Given two sides with lengths 4 and 9, let's denote the length of the third side as \( x \). The triangle inequalities can be expressed as follows:
1. \( 4 + 9 > x \) (which simplifies to \( x < 13 \))
2. \( 4 + x > 9 \) (which simplifies to \( x > 5 \))
3. \( 9 + x > 4 \) (which simplifies to \( x > -5 \), but this condition is always satisfied since \( x \) is a length)
Thus, combining the relevant inequalities from the first two conditions, we find:
\[
5 < x < 13
\]
This means that the third side \( x \) can be any length greater than 5 and less than 13.