In triangle QRS, QR = 8 and RS = 5. Which expresses all possible lengths of side QS? O QS = 13 O 5 < QS < 8 O QS > 13 O 3 < QS < 13
The Correct Answer and Explanation is:
The correct answer is: 3 < QS < 13
Explanation:
In any triangle, the lengths of the sides must satisfy the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For triangle QRS, with sides QR = 8, RS = 5, and QS being the unknown side, we can apply the triangle inequality in the following way:
- QR + RS > QS
This translates to:
8 + 5 > QS
So,
QS < 13 - QR + QS > RS
This translates to:
8 + QS > 5
So,
QS > -3 (which is always true because a side length cannot be negative). - RS + QS > QR
This translates to:
5 + QS > 8
So,
QS > 3
Combining these inequalities:
- From the first inequality, we have QS < 13.
- From the third inequality, we have QS > 3.
Therefore, the possible lengths of side QS must be between 3 and 13, exclusively. This gives us the final range for QS: 3 < QS < 13.
