Question

1. in the figures below, identify as many examples of the terms listed as possible. use symbols to list the examples next to each term.

a. chord:
b. radius:
c. diameter
d. tangent:
e. arc:
f. major arc:
g. minor arc:
h. semicircle:
i. secant:

Explanation:

To solve this, we analyze each term using circle definitions:

a. Chord

A chord is a line segment with both endpoints on the circle.

• $\overline{RS}$: Endpoints $R, S$ on circle $E$.
• $\overline{PK}$: Endpoints $P, K$ on circle $J$.
• $\overline{HT}$: Endpoints $H, T$ on circle $G$.
• $\overline{NU}$: Endpoints $N, U$ on circle $L$.

b. Radius

A radius is a line segment from the center to the circle.

• $\overline{RE}$: Center $E$ to $R$ on circle $E$.
• $\overline{ES}$: Center $E$ to $S$ on circle $E$.
• $\overline{JP}$: Center $J$ to $P$ on circle $J$.
• $\overline{JK}$: Center $J$ to $K$ on circle $J$.
• $\overline{CD}$: Center $C$ to $D$ on circle $C$.

c. Diameter

A diameter is a chord passing through the center (longest chord).

• $\overline{RS}$: Passes through $E$ (center of circle $E$).
• $\overline{PK}$: Passes through $J$ (center of circle $J$).

d. Tangent

A tangent is a line touching the circle at exactly one point.

• Line $WV$: Touches circle $G$ at $V$ (only one point).

e. Arc

An arc is a portion of the circle’s circumference.

• $\overset{\frown}{RZ}$, $\overset{\frown}{ZS}$, $\overset{\frown}{RZS}$ (semicircle), $\overset{\frown}{PK}$, $\overset{\frown}{KQ}$, $\overset{\frown}{QH}$, $\overset{\frown}{HT}$, $\overset{\frown}{TN}$, $\overset{\frown}{NM}$, $\overset{\frown}{MU}$, etc.

f. Major Arc

A major arc is longer than a semicircle (measures $> 180^\circ$).

• $\overset{\frown}{RZS}$ (wait, no—correction: For circle $G$, $\overset{\frown}{HTV}$ (longer than semicircle). For circle $E$, $\overset{\frown}{RZS}$ is a semicircle, so major arcs like $\overset{\frown}{RZS}$ (no, semicircle is $180^\circ$). Correct major arcs: $\overset{\frown}{TVH}$ (circle $G$), $\overset{\frown}{VHT}$ (circle $G$), $\overset{\frown}{RZS}$ is semicircle, so major arcs must be $> 180^\circ$.

g. Minor Arc

A minor arc is shorter than a semicircle (measures $< 180^\circ$).

• $\overset{\frown}{RZ}$, $\overset{\frown}{ZS}$, $\overset{\frown}{PK}$, $\overset{\frown}{KQ}$, $\overset{\frown}{HT}$, $\overset{\frown}{TN}$, $\overset{\frown}{NM}$, $\overset{\frown}{MU}$.

h. Semicircle

A semicircle is an arc measuring $180^\circ$ (half the circle).

• $\overset{\frown}{RZS}$ (circle $E$, since $RS$ is diameter).
• $\overset{\frown}{PKQ}$? No, $\overset{\frown}{PK}$ is diameter, so $\overset{\frown}{PKQ}$? Wait, $\overset{\frown}{RZS}$ (circle $E$) and $\overset{\frown}{PK}$ (diameter, so semicircle $\overset{\frown}{PKQ}$? No, diameter divides circle into two semicircles. So $\overset{\frown}{RZS}$ (circle $E$) and $\overset{\frown}{PK}$ (circle $J$) as semicircles.

i. Secant

A secant is a line intersecting the circle at two points.

• Line $NU$: Intersects circle $L$ at $N$ and $U$.

Final Answers (Key Examples):

a. $\boldsymbol{\overline{RS}, \overline{PK}, \overline{HT}, \overline{NU}}$
b. $\boldsymbol{\overline{RE}, \overline{ES}, \overline{JP}, \overline{JK}, \overline{CD}}$
c. $\boldsymbol{\overline{RS}, \overline{PK}}$
d. $\boldsymbol{WV}$ (line)
e. $\boldsymbol{\overset{\frown}{RZ}, \overset{\frown}{ZS}, \overset{\frown}{PK}, \overset{\frown}{KQ}}$ (examples)
f. $\boldsymbol{\overset{\frown}{TVH}, \overset{\frown}{VHT}}$ (examples)
g. $\boldsymbol{\overset{\frown}{RZ}, \overset{\frown}{ZS}, \overset{\frown}{PK}, \overset{\frown}{KQ}}$ (examples)
h. $\boldsymbol{\overset{\frown}{RZS}, \overset{\frown}{PK}}$ (semicircles)
i. $\boldsymbol{NU}$ (line)

Answer:

To solve this, we analyze each term using circle definitions:

a. Chord

A chord is a line segment with both endpoints on the circle.

• $\overline{RS}$: Endpoints $R, S$ on circle $E$.
• $\overline{PK}$: Endpoints $P, K$ on circle $J$.
• $\overline{HT}$: Endpoints $H, T$ on circle $G$.
• $\overline{NU}$: Endpoints $N, U$ on circle $L$.

b. Radius

A radius is a line segment from the center to the circle.

• $\overline{RE}$: Center $E$ to $R$ on circle $E$.
• $\overline{ES}$: Center $E$ to $S$ on circle $E$.
• $\overline{JP}$: Center $J$ to $P$ on circle $J$.
• $\overline{JK}$: Center $J$ to $K$ on circle $J$.
• $\overline{CD}$: Center $C$ to $D$ on circle $C$.

c. Diameter

A diameter is a chord passing through the center (longest chord).

• $\overline{RS}$: Passes through $E$ (center of circle $E$).
• $\overline{PK}$: Passes through $J$ (center of circle $J$).

d. Tangent

A tangent is a line touching the circle at exactly one point.

• Line $WV$: Touches circle $G$ at $V$ (only one point).

e. Arc

An arc is a portion of the circle’s circumference.

• $\overset{\frown}{RZ}$, $\overset{\frown}{ZS}$, $\overset{\frown}{RZS}$ (semicircle), $\overset{\frown}{PK}$, $\overset{\frown}{KQ}$, $\overset{\frown}{QH}$, $\overset{\frown}{HT}$, $\overset{\frown}{TN}$, $\overset{\frown}{NM}$, $\overset{\frown}{MU}$, etc.

f. Major Arc

A major arc is longer than a semicircle (measures $> 180^\circ$).

• $\overset{\frown}{RZS}$ (wait, no—correction: For circle $G$, $\overset{\frown}{HTV}$ (longer than semicircle). For circle $E$, $\overset{\frown}{RZS}$ is a semicircle, so major arcs like $\overset{\frown}{RZS}$ (no, semicircle is $180^\circ$). Correct major arcs: $\overset{\frown}{TVH}$ (circle $G$), $\overset{\frown}{VHT}$ (circle $G$), $\overset{\frown}{RZS}$ is semicircle, so major arcs must be $> 180^\circ$.

g. Minor Arc

A minor arc is shorter than a semicircle (measures $< 180^\circ$).

• $\overset{\frown}{RZ}$, $\overset{\frown}{ZS}$, $\overset{\frown}{PK}$, $\overset{\frown}{KQ}$, $\overset{\frown}{HT}$, $\overset{\frown}{TN}$, $\overset{\frown}{NM}$, $\overset{\frown}{MU}$.

h. Semicircle

A semicircle is an arc measuring $180^\circ$ (half the circle).

• $\overset{\frown}{RZS}$ (circle $E$, since $RS$ is diameter).
• $\overset{\frown}{PKQ}$? No, $\overset{\frown}{PK}$ is diameter, so $\overset{\frown}{PKQ}$? Wait, $\overset{\frown}{RZS}$ (circle $E$) and $\overset{\frown}{PK}$ (diameter, so semicircle $\overset{\frown}{PKQ}$? No, diameter divides circle into two semicircles. So $\overset{\frown}{RZS}$ (circle $E$) and $\overset{\frown}{PK}$ (circle $J$) as semicircles.

i. Secant

A secant is a line intersecting the circle at two points.

• Line $NU$: Intersects circle $L$ at $N$ and $U$.

Final Answers (Key Examples):

a. $\boldsymbol{\overline{RS}, \overline{PK}, \overline{HT}, \overline{NU}}$
b. $\boldsymbol{\overline{RE}, \overline{ES}, \overline{JP}, \overline{JK}, \overline{CD}}$
c. $\boldsymbol{\overline{RS}, \overline{PK}}$
d. $\boldsymbol{WV}$ (line)
e. $\boldsymbol{\overset{\frown}{RZ}, \overset{\frown}{ZS}, \overset{\frown}{PK}, \overset{\frown}{KQ}}$ (examples)
f. $\boldsymbol{\overset{\frown}{TVH}, \overset{\frown}{VHT}}$ (examples)
g. $\boldsymbol{\overset{\frown}{RZ}, \overset{\frown}{ZS}, \overset{\frown}{PK}, \overset{\frown}{KQ}}$ (examples)
h. $\boldsymbol{\overset{\frown}{RZS}, \overset{\frown}{PK}}$ (semicircles)
i. $\boldsymbol{NU}$ (line)