Question

In circle E, mangle, F, E, G, equals, 50, degrees∠FEG=50 ∘ and the area of shaded sector = start fraction, 5, divided by, 4, end fraction, pi 4/5 π. Find the length of arc, FG ⌢ . Express your answer as a fraction times piπ

Answer 1

The length of the arc FG is
`\( (1)/(3)\pi \)`.

In a circle, the area of a sector can be related to the area of the whole circle using the formula:

`\[ \text{Area of sector} = \left( \frac{\text{measure of central angle}}{360^\circ} \right) * \pi r^2 \]`

where r is the radius of the circle.

In this case, you are given that
`\( \angle FEG = 50^\circ \)` and the area of the shaded sector is
`\( (5)/(4)\pi \)`. Let r be the radius of the circle.

`\[ \text{Area of sector} = (50^\circ)/(360^\circ) * \pi r^2 = (5)/(4)\pi \]`

Now, solve for r :

`\[ (50)/(360) * \pi r^2 = (5)/(4)\pi \]`

Multiply both sides by
`\( (360)/(50) \)` to solve for
`\( r^2 \)`:

`\[ \pi r^2 = (5)/(4) * (360)/(50) * \pi \]`

Simplify:

`\[ r^2 = (9)/(2) \]`

Now, find the length of the arc FG. The formula for the length of an arc, given the central angle in degrees
`(\( \theta \))` and the radius r, is:

`\[ \text{Arc Length} = \left( (\theta)/(360^\circ) \right) * 2\pi r \]`

In this case,
`\( \theta = 50^\circ \)` and
`\( r^2 = (9)/(2) \)`, so
`\( r = \sqrt{(9)/(2)} \)`.

`\[ \text{Arc Length} = (50)/(360) * 2\pi \sqrt{(9)/(2)} \]`

Simplify:

`\[ \text{Arc Length} = (1)/(9) \pi √(18) \]`

Expressing this in the form of a fraction times
`\( \pi \)`:

`\[ \text{Arc Length} = (1)/(9) * 3\pi = (1)/(3)\pi \]`

Therefore, the length of the arc FG is
`\( (1)/(3)\pi \)`.