Question

question 9 of 10
refer to the figure below and mark all statements that are true.

a. if the measure of ( \theta ) is 1 radian, then the arc length is ( 2r ).
b. if the measure of ( \theta ) is ( 60^circ ), then the arc length is ( r ).
c. the ratio of arc length to ( r ) is always equal to ( pi ).
d. if the ratio of the arc length to ( r ) is 1, then the measure of ( \theta ) is 1 radian.

Explanation:

To solve this, we use the formula for arc length \( s = r\theta \), where \( \theta \) is in radians.

Analyzing Each Option:

• Option A:

The arc length formula is \( s = r\theta \). If \( \theta = 1 \) radian, then \( s = r(1) = r \), not \( 2r \). So A is false.

• Option B:

First, convert \( 60^\circ \) to radians: \( 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians.
Using \( s = r\theta \), \( s = r \times \frac{\pi}{3} \approx 1.047r \), not \( r \). So B is false.

• Option C:

From \( s = r\theta \), the ratio \( \frac{s}{r} = \theta \) (in radians). \( \theta \) is only \( \pi \) for a semicircle (\( 180^\circ \)), not always. So C is false.

• Option D:

Given \( \frac{s}{r} = 1 \), from \( s = r\theta \), we divide both sides by \( r \): \( \frac{s}{r} = \theta \). Thus, \( \theta = 1 \) radian. So D is true.

• Option A: Arc length \( s = r\theta \), so \( \theta = 1 \) gives \( s = r \), not \( 2r \). False.
• Option B: \( 60^\circ = \frac{\pi}{3} \) radians, so \( s = r \cdot \frac{\pi}{3}

eq r \). False.

• Option C: \( \frac{s}{r} = \theta \) (radians), not always \( \pi \). False.
• Option D: \( \frac{s}{r} = 1 \implies \theta = 1 \) radian (from \( s = r\theta \)). True.

Answer:

D. If the ratio of the arc length to \( r \) is 1, then the measure of \( \theta \) is 1 radian.