Question

using circle t and circle x, explain the relationship between an arc length and a radius in relation to the central angle that intercepts the arc. move one answer to each box to complete the sentences. the radian measure of a central angle is defined as the ratio of the length of the intercepted arc to the circles radius. thus, we have m∠stu = and m∠wxy = given that all circles are similar and both central angles have the same measure θ, we can express this relationship as the proportion the proportion demonstrates that the length of the intercepted arc is directly proportional

Explanation:

Step1: Recall radian - measure formula

The radian measure $\theta$ of a central angle is defined as $\theta=\frac{s}{r}$, where $s$ is the length of the intercepted arc and $r$ is the radius of the circle.

Step2: Consider similar - circles property

Since all circles are similar and for two circles with central angles $\angle STU$ and $\angle WXY$ having the same measure, if we let $s_1$ and $r_1$ be the arc - length and radius of circle $T$, and $s_2$ and $r_2$ be the arc - length and radius of circle $X$, we have $\frac{s_1}{r_1}=\frac{s_2}{r_2}$.

Answer:

The radian measure of a central angle is defined as the ratio of the length of the intercepted arc to the circle's radius. Thus, we have $\frac{\text{length of arc }ST}{\text{radius of circle }T}=\frac{\text{length of arc }WX}{\text{radius of circle }X}$ and $m\angle STU = m\angle WXY$.