In triangle QRS, QR = 8 and RS = 5. Which expresses all possible length of side QS? A) QS = 13 B) 5 < QS < 8 C) QS > 13 D) 3 < QS < 13
The Correct Answer and Explanation is:
To determine the possible lengths of side QS in triangle QRS, we apply the triangle inequality theorem. The triangle inequality 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.
We know that:
- QR = 8
- RS = 5
- QS is the unknown side.
Applying the triangle inequality theorem:
- QR + RS > QS
This inequality implies:
8+5>QS8 + 5 > QS8+5>QS
13>QS13 > QS13>QS
Hence, QS must be less than 13. - QR + QS > RS
This inequality implies:
8+QS>58 + QS > 58+QS>5
QS>5−8QS > 5 – 8QS>5−8
QS>−3QS > -3QS>−3
Since the length of a side must always be positive, we can disregard the negative result. Therefore, this inequality does not provide new information. - RS + QS > QR
This inequality implies:
5+QS>85 + QS > 85+QS>8
QS>8−5QS > 8 – 5QS>8−5
QS>3QS > 3QS>3
Hence, QS must be greater than 3.
Combining the results:
- From the first inequality, we have QS<13QS < 13QS<13.
- From the second inequality, we have QS>3QS > 3QS>3.
Thus, the possible length of QS lies between 3 and 13, so we have:
3 < QS < 13
Correct Answer:
D) 3 < QS < 13
