LemonmathSelf-study sheet

Grade 9 · Lesson 5.5

Relative positions of two circles

Name: Date: Time spent: min

Goals

Find the relative position of two circles by comparing $OO'$ with $R+r$ and $R-r$ .
Know that the line of centres is a line of symmetry, the perpendicular bisector of the common chord, and passes through the point of contact.
Find the length of a common chord, a radius or the distance between the centres.

Remember

Intersect: $R-r<d<R+r$ ; touch externally: $d=R+r$ ; touch internally: $d=R-r$ ; outside each other: $d>R+r$ ; one inside the other: $d<R-r$ .
The line of centres is a line of symmetry, the perpendicular bisector of the common chord, and passes through the point of contact.
For a common chord: let $OH=x$ and write $AH^2$ in two ways.

Exercises

Exercise 1

Consider the circles $(O, 6\text{ cm})$ and $(O', 4\text{ cm})$ . Find their relative position when $OO'$ is: a) $11$ cm; b) $10$ cm; c) $5$ cm; d) $2$ cm; e) $1$ cm.

Exercise 2

The circles $(O, 5\text{ cm})$ and $(O', 5\text{ cm})$ intersect at $A$ and $B$ , with $OO'=8$ cm. Find $AB$ .

Exercise 3

The circle $(O, 9\text{ cm})$ touches the circle $(O', r)$ internally, with $r<9$ cm and $OO'=4$ cm. Find $r$ .

Exercise 4

The circles $(O, 7\text{ cm})$ and $(O', 3\text{ cm})$ touch each other. Find $OO'$ .

Exercise 5

A logo is made of two circles of radii $3$ cm and $4$ cm whose centres are $5$ cm apart. Minh needs to draw the common chord of the two circles. Find the length of the common chord. (Revise equations reducible to linear equations.)

Exercise 6

Three circles with centres $A$ , $B$ , $C$ touch one another externally in pairs, with $AB=7$ cm, $BC=8$ cm and $CA=9$ cm. Find the radius of each circle. (Revise solving simultaneous equations.)

Exercise 7

(1.5 marks) The circles $(O, 4\text{ cm})$ and $(O', 2\text{ cm})$ touch externally at $A$ . The line $a$ passes through $A$ and is perpendicular to $OO'$ .

a) Find $OO'$ .

b) Prove that $a$ is a common tangent of the two circles.

c) Take a point $M$ on $a$ with $MA=3$ cm. Find $MO$ and $MO'$ (give your answers to 2 decimal places).

Exercise 8 Challenge

The circles $(O, 9\text{ cm})$ and $(O', 4\text{ cm})$ touch externally. A common tangent touches $(O)$ at $B$ and $(O')$ at $C$ , and does not pass through the point where the circles touch. Find $BC$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Consider the circles

(O, 8\text{ cm})

and

(O', 3\text{ cm})

with

OO'=5

cm. The relative position of the two circles is that they

A. intersect.B. touch externally.C. touch internally.D. lie one inside the other.

Question 2. Consider the circles

(O, 6\text{ cm})

and

(O', 2\text{ cm})

with

OO'=3

cm. The relative position of the two circles is that they

A. intersect.B. touch internally.C. lie outside each other.D. lie one inside the other.

Question 3. The circles

(O, 5\text{ cm})

and

(O', 5\text{ cm})

intersect at

A

and

B

, with

OO'=6

cm. The length of the common chord

AB

4

cm.B.

8

cm.C.

6

cm.D.

2\sqrt{34}

cm.

Question 4. The circles

(O, 7\text{ cm})

and

(O', 2\text{ cm})

touch externally. The distance

OO'

5

cm.B.

9

cm.C.

14

cm.D.

4.5

cm.

Part II. True or false

Question 1. Consider the circles

(O, 5\text{ cm})

and

(O', 3\text{ cm})

with

OO'=4

cm. Decide whether each statement is true or false.

a) The two circles intersect.	T / F
b) The line of centres $OO'$ is perpendicular to the common chord.	T / F
c) The two circles touch internally.	T / F
d) The common chord has length $6$ cm.	T / F

Part III. Short answer

Question 1. The circle

(O, 10\text{ cm})

touches the circle

(O', r)

internally, with

r<10

cm and

OO'=3.5

cm. Find

r

(in cm).

Question 2. The circles

(O, 8\text{ cm})

and

(O', 2\text{ cm})

touch externally. A common tangent touches the two circles at

B

and

C

and does not pass through the point where the circles touch. Find

BC

(in cm).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: compare $d$ with $R+r=10$ and $R-r=2$ .

Steps:

Compute $R+r$ and $R-r$ , then compare each value with them.

Exercise 2.

What you need: $OO'$ is the perpendicular bisector of $AB$ ; $H$ is the midpoint of $OO'$ .

Steps:

Find $OH=4$ cm, then $AH$ by Pythagoras' theorem.

Exercise 3.

What you need: touching internally means $d=R-r$ .

Steps:

Solve the equation $9-r=4$ .

Exercise 4.

What you need: touching externally: $d=R+r$ ; touching internally: $d=R-r$ .

Steps:

Look at the two cases.

Exercise 5.

What you need: let $OH=x$ and write $AH^2$ in two ways.

Steps:

Form the equation $3^2-x^2=4^2-(5-x)^2$ .
Find $AH$ , then $AB=2AH$ .

Exercise 6.

What you need: when two circles touch externally, the distance between the centres is the sum of the radii.

Steps:

Form three equations in the unknowns $a$ , $b$ , $c$ , then add them.

Exercise 7.

What you need: the point of contact lies on the line of centres; the test for a tangent.

Steps:

Use $OO'=R+r$ .
Show that $a\perp OA$ and $a\perp O'A$ at $A$ .
Use Pythagoras' theorem in the triangles right-angled at $A$ .

Exercise 8.

What you need: $OB\perp BC$ and $O'C\perp BC$ (property of a tangent).

Steps:

Draw $O'K\perp OB$ at $K$ ; show that $BCO'K$ is a rectangle.
Find $OK$ and $OO'$ , then $O'K$ by Pythagoras' theorem.

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