- How are a circle’s radius and diameter related? - Which circle on a globe passes through the globe’s center? - Think about how to move from diameter to radius and from radius to diameter.

1. The radius is half the diameter, so \(24\,\text{cm} \div 2 = 12\,\text{cm}\). 2. The equator passes through the center of the globe, so its radius is the same as the globe’s radius: \(12\,\text{cm}\). 3. A diameter is twice the radius, so \(2 \cdot 3\,\text{cm} = 6\,\text{cm}\).

a) The radius of the globe is \(12\,\text{cm}\). b) The radius of the equator is \(12\,\text{cm}\). c) The diameter of the circle of latitude is \(6\,\text{cm}\).

- How many \(36^\circ\) angles fit into a full \(360^\circ\) turn? - Use the relationship between a regular polygon's number of sides and its central angle.

1. The central angles of a regular polygon sum to \(360^\circ\). 2. Divide the full angle by one central angle: \(n=\frac{360^\circ}{36^\circ}=10\).

The polygon has \(10\) sides.

- Count the equal sectors in the picture. - How many degrees are in a full turn? - Share that full angle equally among the sectors.

1. The picture shows \(10\) equal slices around the center. 2. The central angles add to \(360^\circ\), so each measures \(360^\circ\div10=36^\circ\).

\(36^\circ\)

- Find the center first. - Check where each segment begins and ends. - Which segment has both endpoints on the circle?

1. Segment \(OA\) joins the center to a point of the circle, so it is a radius. 2. Segment \(BC\) joins two points on the circle, so it is a chord. 3. A radius starts at the center; a chord has both endpoints on the circle.

\(OA\) is a radius, and \(BC\) is a chord. A radius goes from the center to the circle; a chord connects two points on the circle.

- The same endpoints can be joined by two different arcs. - How far around the circle does the route through \(C\) travel?

1. The path starts at \(A\), goes through \(C\), and ends at \(B\), so its name is \(\widehat{ACB}\). 2. It goes more than halfway around the circle, so it is a major arc.

\(\widehat{ACB}\), a major arc.

- Compare the positions of the angle vertices. - Which angle is formed by two radii?

1. Angle \(AOB\) is central because its vertex \(O\) is the center. 2. Angle \(ACB\) is inscribed because its vertex \(C\) is on the circle and its sides are chords.

Central: \(\angle AOB\). Inscribed: \(\angle ACB\). Their vertices lie at the center and on the circle, respectively.

- Every diameter is a chord. What extra condition does it satisfy? - Trace each segment relative to the center. - Length alone is not needed here.

1. Segment \(AB\) has its endpoints on the circle and passes through center \(O\). 2. Therefore, \(AB\) is a diameter as well as a chord. 3. Segment \(CD\) does not pass through \(O\), so it is not a diameter.

\(\overline{AB}\) is a diameter because it passes through the circle's center. \(\overline{CD}\) is a chord but not a diameter.

- How many degrees are in a full circle? - What is true about all central angles of a regular polygon? - For part b, think about how many equal \(24^\circ\) central angles make one full turn.

1. The central angles around the center total \(360^\circ\). 2. For part a, \(360^\circ \div 12 = 30^\circ\). 3. For part b, the number of sides is \(360^\circ \div 24^\circ = 15\).

a) \(30^\circ\) b) \(15\) sides

- Interpret the phrase “at most.” - Decide whether interior points satisfy each condition. - Distinguish the boundary from the boundary together with the interior.

1. The points exactly \(4\,\text{cm}\) from \(M\) form a circle. This set contains only the boundary. 2. The points at most \(4\,\text{cm}\) from \(M\) form a closed disk. This set contains the circle and every point inside it. 3. Therefore, the circle is only the boundary, while the closed disk includes both the boundary and the interior.

The points exactly \(4\,\text{cm}\) from \(M\) form a circle. The points at most \(4\,\text{cm}\) from \(M\) form a closed disk, which includes the circle and its entire interior.

- Find the distance between \(C\) and \(D\). - Recall the relationship between the center distance and radii for externally tangent circles. - Imagine increasing the second radius until the circles just touch.

1. The centers have the same x-coordinate, so \(CD = 7 - 1 = 6\,\text{cm}\). 2. For external tangency, the distance between the centers equals the sum of the radii: \(r_C + r_D = 6\,\text{cm}\). 3. Therefore, \(r_D = 6\,\text{cm} - 3\,\text{cm} = 3\,\text{cm}\).

\(r_D = 3\,\text{cm}\)

- Picture the centers and touching point on one straight line. - How do the radii fit together when the circles touch from the outside? - What changes when the smaller circle touches the inside of the larger one?

1. External tangency means the radii meet end to end, so \(d=2+5=7\,\text{cm}\). 2. For internal tangency, the center distance is the difference between the radii: \(d=5-2=3\,\text{cm}\).

a) \(7\,\text{cm}\) b) \(3\,\text{cm}\)

- Compare the circles using the same measurement: either two radii or two diameters. - How do you find a radius from a diameter? - Compare the two radii after converting.

1. Find the radius of the Oslo circle: \(r_{\text{Oslo}} = 10\,\text{cm} \div 2 = 5\,\text{cm}\). 2. The radius of the Cairo circle is \(8.5\,\text{cm}\). 3. Because \(8.5\,\text{cm} > 5\,\text{cm}\), the circle of latitude through Cairo has the greater radius.

The circle of latitude through Cairo has the greater radius. The Oslo circle has radius \(5\,\text{cm}\), while the Cairo circle has radius \(8.5\,\text{cm}\).

- What does a radius measure, and what does a diameter measure? - How are radius and diameter related? - Convert both measurements to radii or both to diameters before comparing.

1. The diameter of Annie’s circle is twice its radius: \(d = 2r = 2 \cdot 6\,\text{cm} = 12\,\text{cm}\). 2. Tom’s circle also has a diameter of \(12\,\text{cm}\). 3. The circles are the same size. Annie compared measurements that represent different parts of a circle.

Annie is not correct. Her circle has diameter \(2 \cdot 6\,\text{cm} = 12\,\text{cm}\), so the two circles are the same size.

- Does a compass opening represent a radius or a diameter? - First find the diameter of the original circle. - Then find the new diameter and convert it to a radius.

1. The first circle has diameter \(d_1 = 2r_1 = 2 \cdot 5\,\text{cm} = 10\,\text{cm}\). 2. The second circle must have diameter \(d_2 = 2d_1 = 2 \cdot 10\,\text{cm} = 20\,\text{cm}\). 3. A compass is set to the radius, so \(r_2 = 20\,\text{cm} \div 2 = 10\,\text{cm}\). 4. Felix is not correct; he should set the compass to \(10\,\text{cm}\).

Felix is not correct. The first diameter is \(10\,\text{cm}\), so the second diameter is \(20\,\text{cm}\). The required compass setting is the radius, \(10\,\text{cm}\).

- How many radii make one diameter? - Express the new diameter in terms of \(r + 5\). - You may also test the claim with a simple numerical example.

1. For every circle, \(d = 2r\). 2. After the change, the new diameter is \(2(r + 5) = 2r + 10\). 3. The original diameter was \(2r\), so the diameter increases by \(10\,\text{cm}\), not \(5\,\text{cm}\). 4. The claim is false.

The claim is false. Since \(d = 2r\), increasing the radius by \(5\,\text{cm}\) increases the diameter by \(2 \cdot 5\,\text{cm} = 10\,\text{cm}\).

- What do \(MA\) and \(MB\) have in common because \(M\) is the center? - After comparing the two radii, use the single pair of congruence marks in the diagram. - What can you conclude about a triangle whose three sides are congruent?

1. Points \(A\) and \(B\) lie on the circle centered at \(M\), so \(MA\) and \(MB\) are radii of the same circle. Therefore, \(MA=MB\). 2. The diagram marks \(AB\) congruent to \(MA\). Thus \(AB=MA=MB\), so triangle \(AMB\) is equilateral. 3. Every angle of an equilateral triangle measures \(60^\circ\), so \(m\angle AMB=60^\circ\).

a) \(MA=MB\) because both segments are radii of the same circle. b) Triangle \(AMB\) is equilateral, and \(m\angle AMB=60^\circ\).

- How are arc lengths related to their central angles in the same circle? - How many total parts are represented by the ratio \(3:4:5\)? - How many degrees correspond to one ratio part?

1. Arc lengths in the same circle are proportional to their central angle measures. 2. The ratio has \(3+4+5=12\) total parts. 3. One part represents \(360^\circ\div12=30^\circ\). 4. The three central angles are \(3\cdot30^\circ=90^\circ\), \(4\cdot30^\circ=120^\circ\), and \(5\cdot30^\circ=150^\circ\). 5. The largest central angle is \(150^\circ\).

\(150^\circ\)

- Which arc lies inside the marked central angle? - How is the degree measure of a central angle related to the arc it intercepts? - How many degrees are in the entire circle?

1. The marked central angle \(\angle AOB\) measures \(128^\circ\). A central angle and its intercepted arc have the same degree measure, so \(m\widehat{AB}=128^\circ\). 2. A full circle measures \(360^\circ\), so the major arc has measure \(360^\circ-128^\circ=232^\circ\). 3. The central angle with sides through \(A\) and \(B\) is \(\angle AOB\).

a) \(128^\circ\) b) \(232^\circ\) c) \(\angle AOB\)

- Which angle has its vertex at the center and endpoints at \(A\) and \(B\)? - How do central-angle and intercepted-arc degree measures compare? - What is the total measure around the circle?

1. A minor arc and the central angle intercepting it have the same degree measure, so \(m\angle AOB=145^\circ\). 2. The two arcs from \(A\) to \(B\) make a full circle. The other arc is \(360^\circ-145^\circ=215^\circ\).

\(m\angle AOB=145^\circ\) because a central angle has the same degree measure as its intercepted arc. The other arc measures \(360^\circ-145^\circ=215^\circ\) because both arcs make one full circle.

- Can you travel clockwise or counterclockwise from the same point? - Identify a point that lies on each route. - Compare each arc with half the circle.

1. Two different paths on the circle connect the same endpoints. 2. The path through \(C\) is \(\widehat{ACB}\); it is the shorter, minor arc. 3. The path through \(D\) is \(\widehat{ADB}\); it is the longer, major arc. 4. Two endpoint letters alone may need clarification when both arcs are being compared.

\(\widehat{ACB}\) is the minor arc; \(\widehat{ADB}\) is the major arc. They share endpoints but follow different parts of the circle.

- How long is the circle's longest chord? - When can a chord be as long as a diameter?

1. The diameter is \(2\cdot4=8\,\text{cm}\), the greatest possible chord length. 2. A chord reaches that length only if it passes through the center. Segment \(PQ\) does not pass through \(O\). 3. Therefore, \(PQ\) is shorter than \(8\,\text{cm}\), so Maya's claim is false.

No. The longest chord is a diameter, \(8\,\text{cm}\) long. Since \(PQ\) does not pass through \(O\), it must be shorter than \(8\,\text{cm}\).

- What does it mean for two circles to be externally tangent? - Compare the center distance with the sum of the radii. - How can moving one center change that distance?

1. The distance between the centers is \(11-2=9\) units. 2. Two circles touch externally when their centers are separated by the sum of their radii, \(4+2=6\) units. 3. Center \(B\) must move left by \(9-6=3\) units.

Move \(B\) \(3\) units left. The centers start \(9\) units apart and must be \(4+2=6\) units apart for external tangency.

- How does the distance between two centers compare with the sum and the difference of the radii when circles cross twice? - Measure the boat's position from the first lighthouse. - Find where the second light's range begins on that route, then compare the two intervals.

1. The radii differ by \(15-12=3\,\text{mi}\) and add to \(12+15=27\,\text{mi}\). Since \(3<20<27\), the two boundary circles cross at two points. Their interiors overlap over the stated open water. 2. Measure position along the route from \(L_1\). Its light reaches positions up to mile \(12\). 3. The light from \(L_2\) reaches back to mile \(20-15=5\). 4. Both lights are visible from mile \(5\) through mile \(12\), a \(7\,\text{mi}\) stretch.

a) The boundary circles cross at two points because \(3<20<27\); yes, some water lies within both ranges. b) From mile \(5\) through mile \(12\) measured from \(L_1\); the overlap is \(7\,\text{mi}\) long.

- Compare the distance between the centers with the two radii. - For the new position, could one circle lie entirely inside the other? - What changes when circles cross rather than just touch?

1. The radii add to \(4+2=6\,\text{cm}\), the distance between the centers. The circles touch externally at one point on \(\overline{MN}\), \(4\,\text{cm}\) from \(M\). 2. At a center distance of \(5\,\text{cm}\), the distance is less than the sum of the radii, \(6\,\text{cm}\), but greater than their difference, \(2\,\text{cm}\). 3. Neither circle contains the other without crossing, and they no longer touch at just one point. Their boundaries intersect at two points.

a) One point on \(\overline{MN}\), \(4\,\text{cm}\) from \(M\) and \(2\,\text{cm}\) from \(N\). b) Two points, because \(2<5<6\) when distances are in centimeters.

- Count how many hexagon central angles separate each selected pair of vertices. - Relate equal central angles to their chords. - Split one central triangle into two right triangles to find the chord length.

1. Consecutive vertices of a regular hexagon are separated by \(60^\circ\) central angles. 2. Each side of \(\triangle ACE\) skips one hexagon vertex, so each subtends a \(120^\circ\) central angle. 3. Equal central angles intercept congruent chords, so \(AC=CE=EA\). Therefore, \(\triangle ACE\) is equilateral. 4. For chord \(\overline{AC}\), the two radii and the chord form an isosceles triangle with sides \(6\), \(6\), and included angle \(120^\circ\). 5. Bisecting that central triangle gives a right triangle with hypotenuse \(6\) and acute angle \(60^\circ\), so half the chord is \(3\sqrt3\,\text{cm}\). 6. Therefore, each side of \(\triangle ACE\) is \(6\sqrt3\,\text{cm}\).

a) The three sides subtend equal \(120^\circ\) central angles, so the three chords are congruent and \(\triangle ACE\) is equilateral. b) \(6\sqrt3\,\text{cm}\)

- What do five consecutive copies add up to? - How does the angle at the center affect the chord opposite it? - What makes a polygon regular?

1. The five angles total \(5\cdot72^\circ=360^\circ\), so the last ray meets the first ray. 2. Each of the five central angles is \(72^\circ\), so they intercept congruent chords: \(AB=BC=CD=DE=EA\). 3. The five vertices are equally spaced around the circle; they form a regular pentagon.

Five angles of \(72^\circ\) total \(360^\circ\). Their equal central angles give five equal chords, so the pentagon is regular.

- What transformation can move one center to the other without changing size? - What transformation changes every distance from a center by the same factor? - Compare the two radii to determine that factor, then generalize it to radii \(r_1\) and \(r_2\).

1. Translate circle \(c_1\) so that its center \(A\) moves to \(B\). Translation preserves its radius, so the translated circle still has radius \(4\,\text{cm}\). 2. Dilate the translated circle about \(B\) by scale factor \(\frac{10}{4}=\frac{5}{2}\). Every distance from \(B\) is multiplied by \(\frac{5}{2}\), so the radius becomes \(10\,\text{cm}\). The image is \(c_2\). 3. In general, for circles with positive radii \(r_1\) and \(r_2\), first translate one center to the other and then dilate about the common center by scale factor \(\frac{r_2}{r_1}\). 4. A translation followed by a dilation is a similarity transformation, so any two circles are similar.

a) Translate \(A\) to \(B\), then dilate about \(B\) by scale factor \(\frac{5}{2}\). b) For radii \(r_1,r_2>0\), translate the centers together and dilate by \(\frac{r_2}{r_1}\); therefore, all circles are similar.