Question

1. which of the following formulas is used to calculate the length of an arc in radians? a. l = 1/2θr² b. l = θr² c. l = θr d. l = 2πθ 8. if the central angle of a sector is π/2 radians and the radius is 8 meters, what is the area of the s a. 8π square meters b. 64π square meters c. 32π square meters d. 16π square meters 9. which tool is essential for constructing a tangent line from a point outside the circle? a. triangle b. protractor c. ruler d. compass

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $L$ of a circle with radius $r$ and central - angle $\theta$ (in radians) is $L = r\theta$.

Step2: Recall sector - area formula

The formula for the area of a sector $A$ of a circle with radius $r$ and central - angle $\theta$ (in radians) is $A=\frac{1}{2}r^{2}\theta$. Given $\theta=\frac{\pi}{2}$ and $r = 8$ meters, then $A=\frac{1}{2}\times8^{2}\times\frac{\pi}{2}=\frac{1}{2}\times64\times\frac{\pi}{2}=16\pi$ square meters.

Step3: Recall construction tool for tangent

To construct a tangent line from a point outside the circle, a compass is essential. We use the compass to find the mid - point of the line segment joining the center of the circle and the external point, and then draw a circle with that mid - point as the center to find the points of tangency.

Answer:

1. c. $L = r\theta$
2. d. $16\pi$ square meters
3. d. Compass