To find the length of arc \( DF \) in circle \( E \), we need to know the radius of the circle and the measure of the central angle corresponding to arc \( DF \).
Since \( \angle DEF = 90^\circ \), this means that the arc \( DF \) subtended by that angle is a quarter of the circle.
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Determine the radius: We know that \( DE = 17 \) units, which is the radius of the circle \( E \). Thus: \[ r = 17 \text{ units} \]
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Calculate the circumference of the circle: The formula for the circumference \( C \) of a circle is given by: \[ C = 2\pi r \] Substituting the value of the radius: \[ C = 2\pi \cdot 17 = 34\pi \]
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Determine the length of arc \( DF \): Since \( \angle DEF = 90^\circ \), which is \( \frac{1}{4} \) of the full circle (360 degrees), the length of arc \( DF \) can be calculated as: \[ \text{Length of arc } DF = \frac{90^\circ}{360^\circ} \cdot C = \frac{1}{4} \cdot 34\pi \] \[ = \frac{34\pi}{4} = \frac{17\pi}{2} \]
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Compute the numerical value: Now we can approximate \( \frac{17\pi}{2} \): \[ \frac{17\pi}{2} \approx \frac{17 \cdot 3.14159}{2} \approx \frac{53.4071}{2} \approx 26.70355 \]
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Round to the nearest hundredth: Finally, rounding to the nearest hundredth gives us: \[ \text{Length of arc } DF \approx 26.70 \]
Thus, the length of arc \( DF \) is approximately \( \boxed{26.70} \) units.