Introducing circles

The PE teacher ties a $5$ m rope to the tree in the middle of the schoolyard, ties a piece of chalk to the other end, pulls the rope tight and walks all the way round to draw a line. From the tree, An walks $3$ m east and then $4$ m north; Binh walks $4$ m east, $4$ m north; Chi walks $2$ m west, $4$ m south.

Who is standing exactly on the line, who is inside the ring, and who is outside?

The circle

Explore: the points exactly $5$ m from the tree

The chalk always stays exactly $5$ m from the foot of the tree $O$ . East and north are perpendicular, so we find each distance by Pythagoras' theorem:

Student	An	Binh	Chi
Distance to $O$	$\sqrt{3^2+4^2}=5$ m	$\sqrt{4^2+4^2}=\sqrt{32}$ m	$\sqrt{2^2+4^2}=\sqrt{20}$ m

An is exactly $5$ m from $O$ , so An is on the line. The line is made of every point exactly $5$ m from $O$ : it is a circle.

A segment joining the centre to a point of the circle is also called a radius; a segment through the centre with both ends on the circle is a diameter (of length $2R$ ). The circle together with the region inside it is a disc.

Position of a point relative to a circle

We compare the distance from the point to the centre with the radius.

Three positions of a point relative to the circle

(O, R)

Back to the opening question: $OB=\sqrt{32}\approx5.66>5$ and $OC=\sqrt{20}\approx4.47<5$ , so Binh is outside the ring and Chi is inside it.

To prove that several points lie on one circle, we find one point at the same distance from all of them.

A point on the circle is exactly $R$ from the centre.

The symmetry of a circle

Fold a paper disc in half along a crease through the centre, and the two halves fit exactly.

Why? If $A'$ is the reflection of $A$ in $O$ , then $OA'=OA=R$ . If $A''$ is the reflection of $A$ in a line $d$ through $O$ , then $d$ is the perpendicular bisector of $AA''$ , so $OA''=OA=R$ . So $A'$ and $A''$ both lie on the circle.

Key terms

Summary

The circle $(O, R)$ is made of the points exactly $R$ from $O$ ; a diameter has length $2R$ .
$M$ lies on, inside or outside $(O, R)$ when $OM=R$ , $OM<R$ or $OM>R$ .
The centre is a centre of symmetry; every line through the centre is a line of symmetry.

Practice

Each exercise comes with what you need and some steps to start. Try it yourself first, then open the solution.

Exercise 1

Consider the circle $(O, 4\text{ cm})$ and three points $A$ , $B$ , $C$ with $OA=3$ cm, $OB=4$ cm and $OC=4.5$ cm. Does each point lie inside, on or outside the circle?

What you need and how to start

What you need: compare the distance from the centre with $R$ .

Steps:

Compare $OA$ , $OB$ , $OC$ with $4$ .

Show solution

We have $OA=3<4$ , $OB=4$ and $OC=4.5>4$ .

So $A$ lies inside, $B$ lies on and $C$ lies outside the circle.

Answer. $A$ inside, $B$ on, $C$ outside the circle.

Exercise 2

The square $ABCD$ has side $4$ cm. Prove that the four vertices of the square lie on one circle. Find the radius of this circle (give your answer to 2 decimal places).

What you need and how to start

What you need: the diagonals of a square are equal and bisect each other.

Steps:

Find $AC$ by Pythagoras' theorem.
Deduce that $OA=OB=OC=OD$ .

Show solution

We have $AC=\sqrt{4^2+4^2}=4\sqrt2$ (cm). Let $O$ be the point where the diagonals meet; then $OA=OB=OC=OD=\dfrac{AC}{2}=2\sqrt2\approx2.83$ (cm).

So the four vertices lie on the circle $(O, 2\sqrt2\text{ cm})$ , with $R\approx2.83$ cm.

Answer. $R=2\sqrt2\approx2.83$ cm.

Exercise 3

Consider the circle $(O, 5\text{ cm})$ and a point $A$ on it. Let $A'$ be the reflection of $A$ in the point $O$ . Prove that $A'$ lies on the circle and find $AA'$ .

What you need and how to start

What you need: $O$ is the midpoint of $AA'$ .

Steps:

Deduce that $OA'=OA$ , then find $AA'$ .

Show solution

Since $O$ is the midpoint of $AA'$ , we have $OA'=OA=5$ cm, so $A'$ lies on the circle. We have $AA'=OA+OA'=10$ cm.

So $A'$ lies on the circle and $AA'=10$ cm.

Answer. $AA'=10$ cm.

Exercise 4

A loudspeaker at $O$ can be heard clearly within a radius of $30$ m. From $O$ , house $H_1$ is $18$ m east and then $24$ m north; house $H_2$ : $20$ m east, $25$ m north; house $H_3$ : $10$ m east, $27$ m north. Which house hears it clearly, which is exactly on the edge, and which is outside the range?

What you need and how to start

What you need: Pythagoras' theorem; compare with $R=30$ .

Steps:

Compute $OH_1$ , $OH_2$ , $OH_3$ and compare them with $30$ .

Show solution

We have

$OH_1=\sqrt{18^2+24^2}=30,\quad OH_2=\sqrt{1025}\approx32.0>30,\quad OH_3=\sqrt{829}\approx28.8<30.$

So $H_3$ hears it clearly, $H_1$ is exactly on the edge and $H_2$ is outside the range.

Answer. $H_3$ hears it clearly; $H_1$ is on the edge; $H_2$ is outside the range.

Exercise 5

Minh cuts a round pizza of radius $15$ cm along a straight line through the centre. Are the two pieces equal? Find the length of the cut.

What you need and how to start

What you need: every line through the centre is a line of symmetry.

Steps:

Use symmetry; the cut is a diameter.

Show solution

The cut passes through the centre, so it is a line of symmetry, and the two pieces fit exactly when folded. The cut is a diameter, of length $30$ cm.

So the two pieces are equal and the cut is $30$ cm long.

Answer. The two pieces are equal; the cut is $30$ cm long.

Exercise 6

Consider the circle $(O, 6\text{ cm})$ and a point $A$ on it. Take a point $M$ such that $\angle AOM=90^\circ$ and $\angle OMA=30^\circ$ . Does $M$ lie inside or outside the circle? (Revise trigonometric ratios of acute angles.)

What you need and how to start

What you need: the tangent of an acute angle is the opposite side divided by the adjacent side.

Steps:

In the triangle $OMA$ , right-angled at $O$ , write $\tan\angle OMA$ .
Find $OM$ and compare it with $6$ .

Show solution

In the triangle $OMA$ , right-angled at $O$ , we have $\tan\angle OMA=\dfrac{OA}{OM}$ , hence

$OM=\frac{6}{\tan30^\circ}=6\sqrt3\approx10.39\ \text{(cm)}>6\ \text{cm}.$

So $M$ lies outside the circle.

Answer. $M$ lies outside the circle ( $OM=6\sqrt3\approx10.39$ cm).

Exercise 7

(1.5 marks) The triangle $ABC$ is right-angled at $A$ , with $AB=6$ cm and $AC=8$ cm. Let $M$ be the midpoint of $BC$ .

a) Find $BC$ .

b) Prove that $A$ , $B$ , $C$ lie on a circle with centre $M$ . Find the radius of this circle.

c) Let $D$ be the reflection of $A$ in $M$ . Prove that $D$ lies on the circle from part b) and that $ABDC$ is a rectangle.

What you need and how to start

What you need: the median to the hypotenuse is half the hypotenuse.

Steps:

Use Pythagoras' theorem, then show that $MA=MB=MC$ .
Look at the diagonals of $ABDC$ .

Show solution

a) By Pythagoras' theorem, $BC=\sqrt{6^2+8^2}=10$ (cm).

b) We have $MB=MC=MA=\dfrac{BC}{2}=5$ cm (property of the median to the hypotenuse). Hence $A$ , $B$ , $C$ lie on the circle $(M, 5\text{ cm})$ .

c) We have $MD=MA=5$ cm ( $D$ is the reflection of $A$ in $M$ ), so $D$ lies on $(M, 5\text{ cm})$ . The quadrilateral $ABDC$ has diagonals $AD=BC$ that bisect each other, so it is a rectangle.

So $BC=10$ cm, $R=5$ cm and $ABDC$ is a rectangle.

Answer. a) $BC=10$ cm; b) $R=5$ cm; c) $D$ lies on $(M, 5\text{ cm})$ and $ABDC$ is a rectangle.

Exercise 8 Challenge

Consider the circle $(O, 5\text{ cm})$ and a point $M$ with $OM=3$ cm. The point $A$ moves round the circle. Find the greatest and the least value of $MA$ .

What you need and how to start

What you need: the triangle inequality $OA-OM<MA<OA+OM$ .

Steps:

Take $A$ not on the line $OM$ .
Look at the two points where $OM$ meets the circle.

Show solution

If $A$ is not on the line $OM$ , the triangle $OMA$ gives $2<MA<8$ . The line $OM$ cuts the circle at $A_1$ and $A_2$ , with $M$ between $O$ and $A_1$ ; then $MA_1=2$ cm and $MA_2=8$ cm.

So the greatest value of $MA$ is $8$ cm ( $A$ at $A_2$ ) and the least is $2$ cm ( $A$ at $A_1$ ).

Answer. The greatest value of $MA$ is $8$ cm and the least is $2$ cm.

English	Tiếng Việt
circle	đường tròn
centre	tâm
radius	bán kính
diameter	đường kính
disc	hình tròn
line of symmetry	trục đối xứng

a) The point $A$ lies on the circle.	True	False
b) The point $B$ lies outside the circle.	True	False
c) $AC=10$ cm.	True	False
d) The point $C$ lies on the circle.	True	False

The circle

Explore: the points exactly $5$ m from the tree

Position of a point relative to a circle

The symmetry of a circle

Key terms

Summary

Practice

Mini test

Part I. Multiple choice

Part II. True or false

Part III. Short answer

Introducing circles

The circle

Explore: the points exactly 555 m from the tree

Position of a point relative to a circle

The symmetry of a circle

Key terms

Summary

Practice

Mini test

Part I. Multiple choice

Part II. True or false

Part III. Short answer

Explore: the points exactly $5$ m from the tree