Question

explore: measurements of inscribed and central angles
measure of \\(\angle i\\) measure of \\(\angle c\\) \\(\frac{(\angle i)}{\angle c}\\)
39.8 79.6 0.5
how much smaller are the inscribed angles compared to the central angles?

Explanation:

Step1: Analyze the ratio

From the table, the ratio of the inscribed angle ($\angle I$) to the central angle ($\angle C$) is given as $0.5$. This means $\angle I = 0.5\times\angle C$, or $\angle C = 2\times\angle I$.

Step2: Determine how much smaller

To find how much smaller the inscribed angle is compared to the central angle, we can see from the ratio that the inscribed angle is half the measure of the central angle. So the inscribed angle is $\frac{1}{2}$ (or 50% smaller) in terms of the multiple, but in terms of the relationship, the inscribed angle is half the central angle, meaning it is $\frac{\angle C - \angle I}{\angle C}=1 - 0.5 = 0.5$ or 50% smaller in relative terms, or in absolute terms, for the given example, $79.6 - 39.8 = 39.8$, which is equal to the measure of the inscribed angle, showing that the inscribed angle is half the central angle. So the key relationship is that the inscribed angle is half the measure of the central angle, so it is half as large (or 50% smaller) compared to the central angle.

Answer:

The inscribed angles are half as large as the central angles (or the measure of an inscribed angle is $\frac{1}{2}$ the measure of its corresponding central angle, so the inscribed angle is half the size of the central angle).