a) Determine the measure of the central angle that is formed by an arc length of 5 cm in a circle with a radius of 2.5cm. Express the measure in both radians and degrees, correct to one decimal place.

b) Determine the arc length of a circle in part a) if the central angle is 200o.

angle=5/2.5=2 radians= 2*180/PI degrees

a) Well, let's clown around with the central angle first! To find the measure in radians, we can use the formula: ๐œƒ = s / r, where ๐œƒ represents the central angle, s is the arc length, and r is the radius. Plugging in the values, we get: ๐œƒ = 5 / 2.5 = 2.0 radians.

To convert this to degrees, we know that there are 2๐œ‹ radians in a circle, which is equal to 360 degrees. So, using some clown calculations, we can find that: ๐œƒ (in degrees) = ๐œƒ (in radians) ร— (360 / 2๐œ‹) = 2.0 ร— (360 / (2๐œ‹)) = 114.6 degrees.

So, the measure of the central angle is 2.0 radians or 114.6 degrees.

b) Now, if we're talking about a central angle of 200 degrees, we can find the arc length by multiplying the central angle (in degrees) by the radius. Yay, multiplication parade time! Arc length = 200 ร— 2.5 = 500 cm.

So, the arc length is 500 cm.

Remember, clowns are always here to add some laughter to math problems!

a) To determine the measure of the central angle, we can use the formula for the relationship between arc length, radius, and central angle:

Arc Length = Radius * Central Angle

Given that the arc length is 5 cm and the radius is 2.5 cm, we can substitute these values into the formula to find the central angle.

5 cm = 2.5 cm * Central Angle

Solving for Central Angle:

Central Angle = 5 cm / 2.5 cm
Central Angle = 2 radians (rounded to one decimal place)

To express the measure in degrees, we can use the conversion factor: 1 radian = 180 degrees / ฯ€.

Central Angle in Degrees = Central Angle in Radians * (180 degrees / ฯ€)
Central Angle in Degrees = 2 radians * (180 degrees / ฯ€)
Central Angle in Degrees โ‰ˆ 114.6 degrees (rounded to one decimal place)

b) To determine the arc length when the central angle is 200 degrees, we can again use the formula for arc length:

Arc Length = Radius * Central Angle

Given that the radius is 2.5 cm and the central angle is 200 degrees, we can substitute these values into the formula to find the arc length.

Arc Length = 2.5 cm * 200 degrees

Since the central angle is given in degrees, no further conversions are needed.

Arc Length = 2.5 cm * 200 degrees
Arc Length = 500 cm

a) To determine the measure of the central angle formed by an arc length of 5 cm in a circle with a radius of 2.5 cm, we can use the formula for arc length:

Arc Length = Radius * Central Angle

Given that the arc length is 5 cm and the radius is 2.5 cm, we can rearrange the formula to solve for the central angle:

Central Angle = Arc Length / Radius

Central Angle = 5 cm / 2.5 cm

Central Angle = 2 radians

To convert this value to degrees, we can use the conversion factor:

1 radian = (180 degrees) / ฯ€

Therefore, the central angle in degrees is:

Central Angle (in degrees) = Central Angle (in radians) * (180 degrees / ฯ€)

Central Angle (in degrees) = 2 radians * (180 degrees / ฯ€)

Central Angle (in degrees) โ‰ˆ 114.6 degrees

So, the measure of the central angle formed by the given arc length in this circle is approximately 2 radians or 114.6 degrees.

b) To determine the arc length of the circle when the central angle is 200 degrees, we can again use the formula for arc length:

Arc Length = Radius * Central Angle

Given that the radius is 2.5 cm and the central angle is 200 degrees, we can substitute these values into the formula:

Arc Length = 2.5 cm * 200 degrees

Now, let's convert the central angle from degrees to radians using the conversion factor:

1 radian = (180 degrees) / ฯ€

Central Angle (in radians) = Central Angle (in degrees) * (ฯ€ / 180 degrees)

Central Angle (in radians) = 200 degrees * (ฯ€ / 180 degrees)

Central Angle (in radians) โ‰ˆ 3.49 radians

Substituting this value back into the formula for arc length:

Arc Length = 2.5 cm * 3.49 radians

Arc Length โ‰ˆ 8.725 cm

So, the arc length of the circle when the central angle is 200 degrees is approximately 8.725 cm.

The of circle of radius 20cm subten an angle of 120degree at the centre. use the value