Question

In quadrilateral $PQRS$, $PQ = 5$ units, $QR = 17$ units, $RS = 5$ units, and $PS = 9$ units. The length of the diagonal $QS$ can be:

• A. $> 10$ and $< 12$
• B. $> 12$ and $< 14$
• C. $> 14$ and $< 16$
• D. $> 16$ and $< 18$
• E. cannot be determined

Hint:

When asked for possible diagonal lengths in quadrilaterals, always use triangle inequality on both triangles formed by the diagonal and then combine the ranges.

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We are given quadrilateral $PQRS$ with the sides: \[ PQ = 5, QR = 17, RS = 5, PS = 9 \] We want to determine the possible range of diagonal $QS$.
Step 2: Break the quadrilateral into triangles.
Diagonal $QS$ splits the quadrilateral into $\triangle PQS$ and $\triangle QRS$.
- In $\triangle PQS$: sides are $PQ = 5$, $PS = 9$, and diagonal $QS$ (unknown).
- In $\triangle QRS$: sides are $QR = 17$, $RS = 5$, and diagonal $QS$ (unknown).
Step 3: Apply triangle inequality to $\triangle PQS$.
The triangle inequality says: \[ |PQ - PS|<QS<PQ + PS \] \[ |5 - 9|<QS<5 + 9 \] \[ 4<QS<14 \] Step 4: Apply triangle inequality to $\triangle QRS$.
\[ |QR - RS|<QS<QR + RS \] \[ |17 - 5|<QS<17 + 5 \] \[ 12<QS<22 \] Step 5: Combine both ranges.
From $\triangle PQS$: $4<QS<14$
From $\triangle QRS$: $12<QS<22$
Taking the intersection: \[ 12<QS<14 \] \[ \boxed{QS \text{ lies between 12 and 14, i.e., Option (B)}} \]