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Answer :
The measure of the central angle whose radii define the arc is approximately [tex]\( 229.3^\circ \)[/tex]. This means that the angle subtended by the arc is approximately [tex]\( 229.3^\circ \)[/tex]
To find the measure of the central angle whose radii define the arc, we can use the formula that relates the length of an arc to the measure of the central angle.
The formula for finding the length of an arc on a circle is given by:
[tex]\[ \text{Arc length} = \frac{\text{Central angle}}{360^\circ} \times 2\pi r \][/tex]
Where:
- [tex]\( \text{Arc length} \)[/tex] is the length of the arc.
- [tex]\( \text{Central angle} \)[/tex] is the measure of the central angle subtended by the arc.
- r is the radius of the circle.
Given that the arc length is 4r and the radius of the circle is 3 centimeters, we can plug these values into the formula:
[tex]\[ 4r = \frac{\text{Central angle}}{360^\circ} \times 2\pi \times 3 \][/tex]
Now, let's solve for the central angle:
[tex]\[ \text{Central angle} = \frac{4r \times 360^\circ}{2\pi r} \][/tex]
[tex]\[ \text{Central angle} = \frac{4 \times 3 \times 360^\circ}{2\pi \times 3} \][/tex]
[tex]\[ \text{Central angle} = \frac{4 \times 360^\circ}{2\pi} \][/tex]
[tex]\[ \text{Central angle} = \frac{1440^\circ}{2\pi} \][/tex]
[tex]\[ \text{Central angle} \approx \frac{1440^\circ}{6.28} \][/tex]
[tex]\[ \text{Central angle} \approx 229.3^\circ \][/tex]
So, the measure of the central angle whose radii define the arc is approximately [tex]\( 229.3^\circ \)[/tex].
This means that the angle subtended by the arc is approximately [tex]\( 229.3^\circ \)[/tex].
The complete question is:
A circle has radius 3 centimeters: Suppose an arc on the circle has length 4r centimeters: What is the measure of the central angle whose radii define the arc?
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Rewritten by : Jeany
To determine the central angle, we use the arc length formula and convert radians to degrees. The central angle is 240 degrees.
To find the measure of the central angle, we can use the formula that relates arc length to the radius and the central angle in radians:
Arc length (s) = rθ, where r is the radius and θ is the central angle in radians.
Given:
- Radius (r) = 3 cm
- Arc length (s) = 4π cm
We need to find θ:
4π = 3θ
Now solve for θ:
θ = 4π / 3
θ = (4/3)π radians
To convert radians to degrees, we use the conversion factor where π radians = 180 degrees:
θ (degrees) = (4/3)π × (180/π)
θ (degrees) = (4/3) × 180
θ (degrees) = 240 degrees
Therefore, the measure of the central angle is 240 degrees.
