To find the length of one side of the rhombus QRST, we can utilize the properties of the diagonals in a rhombus. The diagonals of a rhombus bisect each other at right angles.
Given:
- Length of diagonal \( QS = 30 \)
- Length of diagonal \( RT = 16 \)
First, we find the lengths of the segments created by the intersection of the diagonals at point U:
- Since diagonal \( QS \) is 30, each half will be: \[ QU = \frac{30}{2} = 15 \]
- Since diagonal \( RT \) is 16, each half will be: \[ RU = \frac{16}{2} = 8 \]
Now, triangle QUR is a right triangle, where:
- \( QU = 15 \)
- \( RU = 8 \)
To find the length of one side of the rhombus (let's call it \( QR \)), we can use the Pythagorean theorem: \[ QR^2 = QU^2 + RU^2 \] Substituting the values we found: \[ QR^2 = 15^2 + 8^2 \] Calculating this: \[ QR^2 = 225 + 64 = 289 \] Taking the square root of both sides to find QR: \[ QR = \sqrt{289} = 17 \]
Therefore, the length of one of the sides of the rhombus is 17.