Question

a circular garden is divided into three sections with different - colored flowers. the measure of ∠bcd is 21°. the measures of bc⏜ and dc⏜ are equal. what is the measure of each arc? enter the correct answer in the box. show hints

Explanation:

Step1: Recall the central - angle - arc relationship

The measure of an arc is equal to the measure of its central angle. Let the measure of $\angle BCD = 21^{\circ}$, and since $\overset{\frown}{BC}=\overset{\frown}{DC}$, and the sum of the measures of the arcs of a circle is $360^{\circ}$. Let the measure of $\overset{\frown}{BC}=x$ and the measure of $\overset{\frown}{DC}=x$, and the measure of the third arc be $y$. So $2x + y=360^{\circ}$. Also, the central angle corresponding to $\overset{\frown}{BC}$ and $\overset{\frown}{DC}$ are equal.

Step2: Calculate the measure of each arc

Since the circle is divided into three sections and $\angle BCD = 21^{\circ}$, and $\overset{\frown}{BC}$ and $\overset{\frown}{DC}$ are equal, the measure of each of $\overset{\frown}{BC}$ and $\overset{\frown}{DC}$ is $179^{\circ}$ and the third arc (the one not equal to the other two) has a measure of $2^{\circ}$. But if we assume that the non - equal part is the angle $\angle BCD = 21^{\circ}$, and the circle is symmetric about the lines dividing the colored sections, and since $\overset{\frown}{BC}=\overset{\frown}{DC}$, we know that the measure of each of $\overset{\frown}{BC}$ and $\overset{\frown}{DC}$ is $\frac{360 - 21}{2}=169.5^{\circ}$

Answer:

$169.5^{\circ}$