## Math 6 Chapter 2 Lesson 8: Circles

## 1. Summary of theory Tóm

### 1.1. Circles and circles

A circle with center O, radius r, denoted (O; r) is a figure consisting of points r at a distance from point O.

**For example: **Circle with center O and radius OM = 1.7 cm.

**Attention:**

For every point in the plane, then:

- If ON < R, then point N is inside the circle (O; R).
- If OM = R, then point M lies on (on) the circle (O;R).
- If OP > R then point M lies outside the circle (O;R).

A circle is a shape consisting of points on a circle and points inside that circle.

### 1.2. Bow – Bow String – Diameter

– Two points A and B lying on the circle divide the circle into two parts, each part is called an arc (arc). Two points A and B are the endpoints of the arc

The line segment AB connecting two ends of the arc is called a chord.

– In case A and B are collinear with O, then each arc is a semicircle

The chord passing through the center is the diameter.

– Diameter is twice as long as radius and is the largest chord.

**For example:** The picture below

- AB and CD are 2 chords.
- The chord AB passes through the center O, so it is also called the diameter AB.
- AB = 2.OA = 2.OB = 2R and \(AB \ge CD\)

## 2. Illustrated exercise

**Question 1:** Let line segment AB = 6cm. Draw a circle with diameter AB and then draw chord CD

**Solution guide**

Let O be the midpoint of line segment AB. The circle with diameter AB is the circle with center O, radius \(R = \frac{{AB}}{2} = 3cm.\)

– Take any 2 points C, D on the circle with center O, we get chord CD.

**Verse 2: **Consider two points A and B that are 4cm apart.

a) On which line are the points 3 cm from A? On which line do the points 2 cm from B lie?

b) Find a point M whose distance to A is 3cm and distance to B is 2cm. How many points are there?

**Solution guide**

a) Points 3cm from A lie on a circle with center A and radius 3cm.

Points 2cm from B lie on a circle with center B and radius 2cm.

b) Points M with MA = 3cm and MB = 2cm are the intersection of 2 circles (A; 3) and (B; 2).

There are 2 points M that satisfy the requirements of the problem.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given a line a and a point \(A \in a\), a length R = 4cm.

a) On which line are the points M in the plane whose distance to point A is 4cm from point A?

b) How many points on the line a are 4 cm from point A? Identify those points.

**Verse 2: **In the plane for a point O. Draw a circle with center O, radius R = 3cm. Given three points \({M_1},{M_2},{M_3}\) knowing \(O{M_1} = 2cm;O{M_2} = 3cm;O{M_3} = 4cm.\) Indicate the positions of the points point \({M_1},{M_2},{M_3}\) for the circle with center O, radius R = 3cm above.

**Question 3: **Consider two points A and B that are 4cm apart.

a) On which line are the points 3 cm from A? On which line are the points 2 cm from B located?

b) Find a point M whose distance to A is 3cm and distance to B is 2cm. How many points are there?

### 3.2. Multiple choice exercises Bài

**Question 1: **On the circle there are 9 distinct points. How many chords are made up of those 9 points?

A. 9

B. 18

C. 72

D. 36

**Verse 2: **How many arcs are made up of 11 distinct points on the circle?

A. 110

B. 120

C. 11

D. 55

**Question 3: **The circle (A; 2cm) and the circle (B; 2cm) intersect at two points C, D. Then the circle (C; 2cm) passes through which of the following points?

A. Grade A

B. DISCLAIMER

C. Both A and B

D. Does not pass through any of the three points A, B, D

**Question 4: **Let the circle (O; 5cm) and OM = 6cm. Select the correct answers:

A. Point M lies on the circle

B. Point M is inside the circle

C. Point M lies outside the circle

D. Point M coincides with the center of the circle

**Question 5: **Given a circle (M; 1.5cm) and three points A, B, C such that OA=1cm, OB=1.5cm, OC=2cm. Select the correct answers.

A. Point A is on the circle, point B is inside the circle and point C is outside of the circle (M; 1.5cm)

B. Point A and Point C lie outside the circle, point B lies on the circle (M; 1.5cm)

C. Point A is inside the circle, point B is on the circle and point C is outside of the circle (M; 1.5cm)

D. All three points lie on the circle (M; 1.5cm)

## 4. Conclusion

Through this lesson, you should understand the following main topics:

- Know the concepts of circle, circle, center, arc, chord, diameter, and radius.
- Identify points inside, above, and outside of a circle.
- Draw a circle with given center and radius. Know the names and symbols of circles.

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