Question

if the central angles of circle e and circle o are congruent, and the length of lt is proportional to el, which statement about the length of jy is true?
the length of jy is proportional to el.
the length of jy is less than oj.
the length of jy is proportional to oj.
the length of jy is less than el.

Explanation:

Step1: Recall arc - length formula

The length of an arc $s = r\theta$ (where $r$ is the radius of the circle and $\theta$ is the central - angle in radians). If two circles are congruent, their radii are equal. Let the radius of circle $E$ and circle $O$ be $r$. For an arc $\overset{\frown}{JY}$ in circle $O$ and an arc $\overset{\frown}{EL}$ in circle $E$, if the central angles corresponding to them are $\theta_1$ and $\theta_2$ respectively, then the lengths of the arcs are $s_{JY}=r\theta_1$ and $s_{EL}=r\theta_2$. The length of an arc is proportional to the measure of its central angle when the radii are equal (since $s = r\theta$ and $r$ is constant for congruent circles).

Step2: Analyze each option

• Option 1: There is no information to suggest that the length of $\overset{\frown}{JY}$ is less than $\overset{\frown}{EL}$ without knowing the central - angle measures.
• Option 2: There is no relation between the length of an arc and a non - related radius ($OJ$) in the sense of proportionality. The length of an arc is proportional to the central angle when the radius is constant (for the same or congruent circles).
• Option 3: There is no information to suggest that the length of $\overset{\frown}{JY}$ is less than $OJ$ as $OJ$ is a radius and arc - length and radius are related by $s = r\theta$ but not in a way of direct comparison of less than for no given $\theta$.
• Option 4: Since the circles are congruent (same radius) and the length of an arc $s=r\theta$ (where $r$ is the radius and $\theta$ is the central angle), the length of an arc is proportional to the central angle.

Answer:

The length of $\overset{\frown}{JY}$ is proportional to $EL$.