Question

type the correct answer in each box. use numerals instead of words.

consider the given circle with the shaded sector xy and central angle, \\(225^{\circ}\\). the circumference of the circle is \\(30\pi\\) units.

use the given information to complete the statements. round any non-integer answers to the hundredths place.

the length of major arc xy is units. the radius of the circle is units. the area of the shaded sector is square units.

Explanation:

⚡ Using what you learned: Parts of a Circle (radius, diameter, chord, tangent, secant, arc, sector)

Step 1: Find the radius of the circle

The circumference \( C \) of a circle is given by:

$$ C = 2\pi r $$

Given that the circumference is \( 30\pi \) units:

$$ 2\pi r = 30\pi $$

$$ r = 15 $$

Step 2: Find the length of major arc XY

The central angle for the shaded major arc \( XY \) is \( 225^\circ \).

The arc length formula is:

$$ \text{Arc Length} = \frac{\theta}{360^\circ} \times C $$

Substitute \( \theta = 225^\circ \) and \( C = 30\pi \):

$$ \text{Arc Length} = \frac{225}{360} \times 30\pi $$

$$ \text{Arc Length} = \frac{5}{8} \times 30\pi = 18.75\pi $$

Convert to a decimal rounded to the hundredths place:

$$ 18.75 \times 3.14159265... \approx 58.90 $$

Step 3: Find the area of the shaded sector

The area of a sector formula is:

$$ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 $$

Substitute \( \theta = 225^\circ \) and \( r = 15 \):

$$ \text{Area} = \frac{225}{360} \times \pi (15)^2 $$

$$ \text{Area} = \frac{5}{8} \times 225\pi = 140.625\pi $$

Convert to a decimal rounded to the hundredths place:

$$ 140.625 \times 3.14159265... \approx 441.79 $$

Answer:

• The length of major arc XY is 58.90 units.
• The radius of the circle is 15 units.
• The area of the shaded sector is 441.79 square units.