Question

types of arcs determine whether the arc is a minor arc, a major arc, or a semicircle of $odot r$. $overline{pt}$ and $overline{qu}$ are diameters.

1. $widehat{pq}$
2. $widehat{su}$
3. $widehat{pqt}$
4. $widehat{qt}$
5. $widehat{tuq}$
6. $widehat{tup}$
7. $widehat{qut}$
8. $widehat{puq}$

Explanation:

Step1: Define arc types

A minor arc is less than $180^\circ$, major arc is more than $180^\circ$, semicircle is exactly $180^\circ$. Diameters split the circle into $180^\circ$ halves, and the circle is split into 4 equal $90^\circ$ arcs here.

Step2: Analyze $\overset{\frown}{PQ}$

$\overset{\frown}{PQ}$ spans 1 quarter ($90^\circ$), minor arc.

Step3: Analyze $\overset{\frown}{SU}$

$\overset{\frown}{SU}$ spans 1 quarter ($90^\circ$), minor arc.

Step4: Analyze $\overset{\frown}{PQT}$

$\overset{\frown}{PQT}$ spans 3 quarters ($270^\circ$), major arc.

Step5: Analyze $\overset{\frown}{QT}$

$\overset{\frown}{QT}$ spans 1 quarter ($90^\circ$), minor arc.

Step6: Analyze $\overset{\frown}{TUQ}$

$\overset{\frown}{TUQ}$ spans 3 quarters ($270^\circ$), major arc.

Step7: Analyze $\overset{\frown}{TUP}$

$\overset{\frown}{TUP}$ spans 2 quarters ($180^\circ$), semicircle.

Step8: Analyze $\overset{\frown}{QUT}$

$\overset{\frown}{QUT}$ spans 3 quarters ($270^\circ$), major arc.

Step9: Analyze $\overset{\frown}{PUQ}$

$\overset{\frown}{PUQ}$ spans 3 quarters ($270^\circ$), major arc.

Answer:

1. Minor arc
2. Minor arc
3. Major arc
4. Minor arc
5. Major arc
6. Semicircle
7. Major arc
8. Major arc