To find the length of one of the sides of rhombus QRST given the lengths of the diagonals QS and RT, we can use the properties of rhombuses.
In a rhombus, the diagonals bisect each other at right angles. This means that point U, where the diagonals intersect, divides QS and RT into two equal halves. Let's find the lengths of the halves of each diagonal:
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Length of diagonal QS is 30, so each half (QU and US) is: \[ QU = US = \frac{30}{2} = 15 \]
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Length of diagonal RT is 16, so each half (RU and UT) is: \[ RU = UT = \frac{16}{2} = 8 \]
Now, we can form a right triangle using half of each diagonal and one side of the rhombus (e.g., side QR). In triangle QUR, we have:
- QU = 15 (one leg)
- RU = 8 (the other leg)
We can use the Pythagorean theorem to find the length of side QR (which is equal to the length of any side of the rhombus):
\[ QR^2 = QU^2 + RU^2 \] \[ QR^2 = 15^2 + 8^2 \] \[ QR^2 = 225 + 64 \] \[ QR^2 = 289 \]
Taking the square root, we find:
\[ QR = \sqrt{289} = 17 \]
Thus, the length of one side of the rhombus is 17.
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