Question

In circle E with m, angle, D, E, F, equals, 90, degreesm∠DEF=90
∘
and D, E, equals, 17DE=17 units, find the length of arc, D, F
DF
⌢
. Round to the nearest hundredth.

Answer 1

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.

1. **Determine the radius**:
We know that \( DE = 17 \) units, which is the radius of the circle \( E \). Thus:
\[
r = 17 \text{ units}
\]

2. **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:
\[
C = 2\pi \cdot 17 = 34\pi
\]

3. **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:
\[
\text{Length of arc } DF = \frac{90^\circ}{360^\circ} \cdot C = \frac{1}{4} \cdot 34\pi
\]
\[
= \frac{34\pi}{4} = \frac{17\pi}{2}
\]

4. **Compute the numerical value**:
Now we can approximate \( \frac{17\pi}{2} \):
\[
\frac{17\pi}{2} \approx \frac{17 \cdot 3.14159}{2} \approx \frac{53.4071}{2} \approx 26.70355
\]

5. **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.