## Math 9 Review Chapter 2: Circles

## 1. Summary of theory

### 1.1. The concept of diameter

Of the strings of a circle, the largest wire is the diameter

In a circle, a diameter perpendicular to a string passes through its midpoint

In a circle, the diameter through the midpoint of a string that does not pass through the center is perpendicular to the string.

### 1.2. Relationship between wire and center-to-wire distance

**Theorem 1:**

In a circle

a) If two strings are equal, they are equidistant from the center

b) Two strings equidistant from the center are equal

**Theorem 2:**

In a circle

a) Whichever string is larger, it is closer to the center

b) The string that is closer to the center is the larger string

### 1.3. The three relative positions of the line and the circle

**a) The line and the circle intersect**

When a line a and a circle (O; R) have 2 points in common, we say that the line a and the circle (O; R) intersect. The line a is called the tangent of the circle (O; R).

Then: Let H be the perpendicular projection of O onto a, then OH is the distance from O to a and OH

**b) The line and the circle touch each other**

When the line a and the circle (O; R) have a common point at C, we say the line a and the circle (O; R) touch each other.

We also say that the line a is a tangent to the circle. Point C is called the contact and OC is the distance from O to a. Then OH = R

**Theorem:**

If a line is a tangent to a circle, then it is perpendicular to the radius passing through the point of contact

**c) The line and the circle do not intersect**

When the line a and the circle (O) have no points in common, we say that the line a and the circle (O) do not intersect.

### 1.4. The relation between the distance from the center of the circle to the line and the radius of the circle

Given a straight line a and (O; R). Let OH = d be the distance from O to the line a. Then:

d < R <=> line a intersects (O; R) at two distinct points

d = R <=> the line a has a point in common with (O; R) (or the line a is tangent to the circle (O; R))

d > R <=> the line a has no point in common with the circle (O; R)

### 1.5. Signs to identify the tangent of a circle

**Theorem:**

If a line passes through a point of a circle and is perpendicular to the radius passing through that point, then the line is a tangent to the circle.

### 1.6. Theorem of two intersecting tangents

**Theorem:**

If two tangents to a circle intersect at a point, then:

– That point is equidistant from the two contacts

– The ray from that point passing through the center is the bisector of the angle formed by the two tangents

– The ray from the center passing through that point is the bisector of the angle formed by the two radii passing through the points of contact

– Angle formed by two tangents AB and AC is angle BAC

– Angle formed by two radii passing through the contacts is BOC

### 1.7. Circle inscribed in triangle

The circle that touches the three sides of a triangle is called the incircle of the triangle, and the triangle is called the circumcircle.

The center of the incircle of a triangle is the intersection of the interior bisectors of that triangle.

### 1.8. Inner line properties

**Theorem:**

– If two circles intersect, then the two intersections are symmetrical through the line joining the center, that is, the line joining the center is the perpendicular bisector of the common chord.

– If two circles touch each other, the point of contact lies on the line connecting the center

### 1.9. The relation between the segment connecting the center and the radii

Consider two circles (O; R) and (O’; r) where \(R \ge r\)

**a) Two circles intersect each other**

If two circles (O; R) and (O’; r) intersect, then R – r < OO' < R + r

**b) Two circles touch each other**

– If (O) and (O’) are in external contact then: OO’ = R + r

– If (O) and (O’) are in contact, then: OO’ = R – r

**c) Two circles do not intersect**

– If two circles (O) and (O’) are outside, then OO’ > R -r

### 1.10. Common tangent to two circles

The common tangent to two circles is the line tangent to both circles

– The common outer tangent does not intersect the interior segment

– Common tangent in the segment joining the center

– If circle (O) contains circle (O’), then OO’ < R - r

– If two circles (O) and (O’) are concentric, then OO’ = 0

## 2. Illustrated exercise

### 2.1. Exercise 1

Let the circle (O;R) and the two chords AB and CD are equal and perpendicular to each other at I. Assume IA=4, IB=8. Distance from center O to AB is d and to CD is d’

Values of d and d’

**Solution guide**

Let E, F be the perpendicular projection of O onto CD, AB respectively. Since quadrilateral OFIE has 3 right angles, OFIE is a rectangle

we have OE=OF because AB=CD, so OFIE is a square then:

\(OE=OF=EI=FI=FA-IA=\frac{AB}{2}-IA=\frac{IA+IB}{2}-IA=2\)

### 2.2. Exercise 2

Given (O;10), string AB=20. Draw string CD parallel to AB and with a distance of 8 from AB. What is the length of cord CD?

**Solution guide**

Since the diameter of the circle is 20, AB passes through the center of the circle.

Let E be the midpoint of CD \(\Rightarrow OE\perp AB\)

In the right triangle OEC at E, we have: \(CE=\sqrt{CO^2-OE^2}=6\)

\(CD=2CE=12\)

### 2.3. Exercise 3

Given a circle (O;3). A point A is 8 distances from O. Draw a tangent to AB with (O) (B is the tangent). The length AB is:

**Solution guide**

Based on the above figure, we see that:

Triangle AOB is right-angled at B

\(\Rightarrow AB=\sqrt{AO^2-OB^2}=\sqrt{55}\)

### 2.4. Exercise 4

Given a circle (O;4). A point A is at a distance of 12 from O. Draw a tangent to AB to (O) (B is the tangent). OA intersects the circle at C. Through C draw a line parallel to OB, intersecting AB at D. What is the magnitude of CD?

**Solution guide**

We have: \(CD//OB\Rightarrow \frac{CD}{OB}=\frac{AC}{AO}\)\(\Rightarrow CD=\frac{AC.OB}{AO}=\frac{ 8.4}{12}=\frac{8}{3}\)

### 2.5. Exercise 5

Let ABC be a right triangle at A with AB=6, AC=8. Circle (I;r) inscribed in triangle ABC. The value of r is:

**Solution guide**

\(S_{ABC}=\frac{1}{2}.AB.AC=24=pr\Rightarrow r=\frac{24}{\frac{1}{2}.(AB+AC+\sqrt{AB) ^2+AC^2})}=2\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Let two circles (O) and (O’) touch the outside at A. Draw the diameters AOB, AO’C. Let DE be the common tangent to the two circles (D ∈ (O), E ∈ (O’)). Let M be the intersection of BD and CE.

a. Calculate the measure of angle DAE.

b. What is the quadrilateral ADME? Why ?

c. Prove that MA is a common tangent to the two circles.

**Verse 2: **Let two circles (O) and (O’) tangent externally at A. Draw a common tangent outside MN of the two circles (M ∈ (O), N ∈ (O’)). Let P be the point of symmetry to M over OO’, Q the point of symmetry to N over OO’. Prove that:

a. MNQP is an isosceles trapezoid.

b. PQ is the common tangent to the two circles (O) and (O’).

c. MN + PQ = MP + NQ.

**Question 3: **Given two circles (O; 2cm), (O’; 3cm), OO’ = 6cm

a. How are the two circles (O) and (O’) relative to each other?

b. Draw a circle (O’; 1cm) and then draw tangent OA to that circle (A is the tangent). Ray O’A intersects the circle (O’; 3cm) at B. Draw the radius OC of the circle (O) parallel to O’B, B and C in the same half-plane with edge OO’. Prove that BC is a common tangent to two circles (O; 2cm), (O’; 3cm).

c. Calculate the length BC

d. Let I be the intersection of BC and OO’/ Calculate the length IO

**Question 4: **Given a circle (O; R), point A lies outside the circle (R < OA < 3R). Draw a circle (A; 2R)

a. How are the two circles (O) and (A) relative to each other?

b. Let B be an intersection of the two circles above. Draw the diameter BOC of the circle (O). Let D be the intersection (other than C) of AC and the circle (O). Prove that AD = DC

### 3.2. Multiple choice exercises

**Question 1: **Let two circles (O;4) and (O’;4) intersect at A and B. Knowing OO’=6. The common chord length AB is:

A. \(AB=\sqrt{7}\)

B. \(AB=2\sqrt{7}\)

C. \(AB=7\)

D. \(AB=14\)

**Verse 2: **Cho (O). From a point M outside (O) two tangents MA, MB such that \(\widehat{AMB}=60^{\circ}\). The perimeter of triangle MAB is 30. Calculate the length of the string AB

A. \(5\)

B. \(5\sqrt{2}\)

C. \(5\sqrt{3}\)

D. \(10\)

**Question 3: **Given circles (O;5) and (O’;4). Knowing OO’=10. The relative positions of the two circles are

A. Do not cut each other

B. Cut each other

C. Exposure in

D. External contact

**Question 4: **Let ABC be a triangle with AB=5, AC=12, BC=13. Then:

A. AB is a tangent to (C;5)

B. AC is tangent to (B;5)

C. AB is tangent to (B;12)

D. AC is tangent to (C;13)

**Question 5: **Given a circle (O;R) and a string CD. From O draw a ray perpendicular to CD at M, intersecting (O) at H (M lies between O and H). Knowing CD=16, MH=4. R=?

A. 10

B. 12

C. 14

D. 16

## 4. Conclusion

Through this lesson, you will learn some key topics as follows:

- Systematize the knowledge about the symmetry of the circle, the relationship between the string and the distance from the center to the string, the relative position of the two circles, the line and the circle.
- Proficient in drawing skills, applying learned knowledge to solve math and proof problems.

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