## Math 9 Chapter 2 Lesson 8: Relative positions of two circles (continued)

## 1. Summary of theory

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

**a) Two circles intersect each other**

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

**b) Two circles touch each other**

– If two circles (O) and (O’) touch externally, then OO’ = R + r

– If two circles (O) and (O’) touch internally, then OO’ = R – r

**c) Two circles do not intersect**

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

– If two circles (O) contain a circle (O’), then OO’ > R – r

### 1.2. Common tangent to two circles

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

## 2. Illustrated exercise

### 2.1. Basic exercises

Given a circle with center O and radius OA and a circle with diameter OA.

a) Determine the relative positions of the two circles

b) The chord AD of the great circle intersects the small circle at C. CMR: AC=CD

**Solution guide**

a) Two circles with center O and radius OA and a circle with diameter OA are internally tangent

b) Triangle AOC has IA=IO=IC, so the triangle is right angled at C or OC is perpendicular AD at C

So C is mid point of AD so AC=CD

### 2.2. Advanced exercises

**Question 1: **Given 2 circles (O;R) and (O;r) intersect at two points A and B. Draw the diameters AOC and AO’D

a) Prove that 3 points C, B, D are collinear

b) Through A draw the intersecting lines (O) and (O’) at M, N respectively. CMR: \(MN\leq CD\)

**Solution guide**

a) Triangle ABC has AC as diameter, so triangle ABC is right angled at B or \(AB\perp CB\)

Triangle ABD has AD as diameter, so triangle ABD is right angled at B or \(AB\perp BD\)

\(\Rightarrow C,B,D\) on the same line through B and perpendicular to AB

b) Consider triangle ACD with OO’ as the median, so: \(OO’=\frac{1}{2}.CD\)

Let E and F be the perpendicular projections of O and O’ onto MN, respectively. Then E and F are midpoints AM and AN respectively

deduce \(EF=\frac{1}{2}.MN\). We take the comparison of CD with MN through comparison of OO’ and EF .

Consider two lines OE and O’F that are parallel to each other. EF is perpendicular to both lines, so EF is the smallest of the segments connecting from a point on OE to a point on O’F.

\(EF\leq OO’\Rightarrow MN\leq CD\)

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Let I be the midpoint of line segment AB. Draw circles (I; IA) and (B; BA).

a. How are the two circles (I) and (B) above relative to each other? Why?

b. Draw a line through A, intersecting the circles (I) and (B) at M and N respectively. Compare the lengths AM and MN.

**Verse 2: **Given two concentric circles O. Let AB be any chord of the small circle. The line AB intersects the great circle at C and D (A lies between B and C). Compare the lengths AC and BD.

**Question 3:** Let two circles (O) and (O’) touch externally at A. Let CD be the common external tangent to the two circles (C ∈ (O), D ∈ (O’))

a. Calculate angle CAD

b. Calculate the length of CD knowing OA = 4.5cm, O’A = 2cm

**Question 4: **Given two concentric circles O. One circle (O’) intersects one circle with center O at A, B and the other with center O at C, D. Prove that AB // CD.

### 3.2. Multiple choice exercises

**Question 1: **Given 2 circles (O;R) and (O’;r), R>r

Which of the following statements is incorrect?

A. Two circles (O) and (O’) intersect if and only if Rr

B. Two circles (O) and (O’) are tangent if and only if OO’=Rr

C. Two circles (O) and (O’) touch while and only if OO’=Rr

D. Two circles (O) and (O’) are said to be outside each other if and only if OO’>R+r

**Verse 2: **Given 3 circles (A), (B), (C) with the same radius R are tangent to each other. Let D, E, F be the contacts. The area of triangle DEF is:

A. \(\frac{R^2\sqrt{3}}{2}\)

B. \(\frac{R^2\sqrt{3}}{3}\)

C. \(\frac{R^2\sqrt{3}}{6}\)

D. \(\frac{R^2\sqrt{3}}{4}\)

**Question 3: **Given circles (O;10) and (O’;3). Knowing OO’=8. The relative positions of the two circles are

A. (O) contains (O’)

B. Cut each other

C. Exposure in

D. External contact

**Question 4: **Let two circles (O;5) and (O’;5) intersect at A and B. Knowing OO’=8. The common chord length AB is:

A. 4

B. 5

C. 6

D. 7

**Question 5:** Given a circle (O;9). Draw 6 equal circles with radius R all tangent to (O) and each tangent to 2 other circles next to it. R=?

A. \(6\)

B. \(3\)

C. \(2\sqrt{3}\)

D. \(3\sqrt{3}\)

## 4. Conclusion

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

- Knowing the three relative positions of two circles, the common tangent between the two circles.
- Apply the properties of two intersecting, tangent circles in the exercise of calculation and proof.

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