To find the value of \( x \) in pentagon PQRST, we first need to determine the scale factor between the two pentagons.
The side lengths of pentagon ABCDE are:
- \( AB = 5 \)
- \( BC = 6 \)
- \( CD = 5 \)
- \( DE = 5 \)
- \( EA = 5 \)
The side lengths of pentagon PQRST are:
- \( PQ = 2.5 \)
- \( QR = x \)
- \( RS = 2.5 \)
- \( ST = 2.5 \)
- \( TP = 2.5 \)
The scale factor from pentagon ABCDE to pentagon PQRST can be determined by comparing the lengths of the corresponding sides.
For example, consider \( AB = 5 \) (from ABCDE) and \( PQ = 2.5 \) (from PQRST): \[ \text{Scale factor} = \frac{\text{Length of PQ}}{\text{Length of AB}} = \frac{2.5}{5} = \frac{1}{2} \]
This scale factor applies to all corresponding sides. Therefore, all sides of pentagon ABCDE are halved to get the corresponding sides of pentagon PQRST.
Now, applying this scale factor to side \( BC \): \[ BC = 6 \implies QR = \frac{1}{2} \times 6 = 3 \]
Thus, the value of \( x \) is: \[ x = 3 \]
So, the answer is: \[ \boxed{3} \]
9 answers