Question

geometry: concepts and connections unit 6 georgias k - 12 mathematics standards
name:
date:
central angles, arcs, and chords
formative assessment question:

1. given the circle, name the following:

a. major arc
b. minor arc
c. central angle
d. radius
e. diameter

1. the ratio of the circumference of circle d to the circumference of circle c is $\frac{3}{2}$. what is the ratio of their areas?
2. the ratio of the area of circle f to the area of circle e is $\frac{25}{16}$. what is the ratio of their radii?

Explanation:

Step1: Identify circle - related terms for question 1

a. A major arc is an arc that is larger than a semi - circle. For example, arc $ABC$.
b. A minor arc is an arc that is smaller than a semi - circle. For example, arc $AC$.
c. A central angle is an angle whose vertex is at the center of the circle. $\angle AMC$ is a central angle.
d. A radius is a line segment from the center of the circle to a point on the circle. $MA$, $MC$, $MB$ are radii.
e. A diameter is a line segment that passes through the center of the circle with both endpoints on the circle. $AB$ is a diameter.

Step2: Solve for ratio of areas in question 2

The formula for the circumference of a circle is $C = 2\pi r$. Let the radius of circle $D$ be $r_D$ and of circle $C$ be $r_C$. Given $\frac{C_D}{C_C}=\frac{2\pi r_D}{2\pi r_C}=\frac{3}{2}$, so $\frac{r_D}{r_C}=\frac{3}{2}$. The formula for the area of a circle is $A=\pi r^{2}$. Then $\frac{A_D}{A_C}=\frac{\pi r_D^{2}}{\pi r_C^{2}}=(\frac{r_D}{r_C})^2 = (\frac{3}{2})^2=\frac{9}{4}$.

Step3: Solve for ratio of radii in question 3

Let the radius of circle $F$ be $r_F$ and of circle $E$ be $r_E$. Given $\frac{A_F}{A_E}=\frac{\pi r_F^{2}}{\pi r_E^{2}}=\frac{25}{16}$. Then $\frac{r_F^{2}}{r_E^{2}}=\frac{25}{16}$, so $\frac{r_F}{r_E}=\sqrt{\frac{25}{16}}=\frac{5}{4}$.

Answer:

1.
a. Example: arc $ABC$
b. Example: arc $AC$
c. $\angle AMC$
d. $MA$, $MC$, $MB$
e. $AB$

1. $\frac{9}{4}$
2. $\frac{5}{4}$